If Christ has all power to perform miracles, how is that that "he could there do no mighty work" save that He healed a few?

The 'couldn't' doesn't denote lack of power, but lack of opportunity. It says this in the verse itself.

Mark 6:5-6 (DRB) And he could not do any miracles there, only that he cured a few that were sick, laying his hands upon them. 6 And he wondered because of their unbelief, and he went through the villages round about teaching.

The people there did not have faith, and so He was not willing to “throw pearls to swine” (Matthew 7:6; Cf. 16:4). Besides, working a miracle for faithless people would not be right, as it makes faith by implication obsolete and valueless, if not in reality, then at least to all that see Him performing signs for those with no faith and those who really have faith—confirms them in the 'okayness' of unbelief.

Even on an atheistic level, the author of Mark (from now on 'Mark' ...), Mark has no reason to arbitrarily show that Jesus arbitrarily couldn't work miracles in a certain geographical location. And it's not coincidental that Jesus is shocked at their great "unbelief." The Jesus that 'cannot' (as in has no power to) work a miracle when He steps foot in a certain place is not the Jesus presented by Mark.

Mark 4:25-40 (DRB)

And he saith to them that day, when evening was come: Let us pass over to the other side. 36 And sending away the multitude, they take him even as he was in the ship: and there were other ships with him. 37 And there arose a great storm of wind, and the waves beat into the ship, so that the ship was filled. 38 And he was in the hinder part of the ship, sleeping upon a pillow; and they awake him, and say to him: Master, doth it not concern thee that we perish? 39 And rising up, he rebuked the wind, and said to the sea: Peace, be still. And the wind ceased: and there was made a great calm. 40 And he said to them: Why are you fearful? have you not faith yet? And they feared exceedingly: and they said one to another: Who is this (thinkest thou) that both wind and sea obey him?

For crying out loud, this is the 'Yahweh' stuff of Psalms:

Psalm 89:9-10 (DRB) O Lord God of hosts, who is like to thee? thou art mighty, O Lord, and thy truth is round about thee. 10 Thou rulest the power of the sea: and appeasest the motion of the waves thereof.

Psalm 106:25-29 (DRB)

He said the word, and there arose a storm of wind: and the waves thereof were lifted up. 26 They mount up to the heavens, and they go down to the depths: their soul pined away with evils. 27 They were troubled, and reeled like a drunken man; and all their wisdom was swallowed up. 28 And they cried to the Lord in their affliction: and he brought them out of their distresses. 29 And he turned the storm into a breeze: and its waves were still.

I gave an answer on a similar question here. My premise is that Christ had a priority on kingdom works, which were as follows:

1. Dying on the cross.
2. Preaching repentance/etc. and loving others as God does.
3. Healing/miracles.

The healings and miracles, per my comments above, were not His main thing. He used miracles to prove His divinity and supernatural nature as both God and man and bring people back to the Father through those.

In the case of Mark 6:5-6, there being no mighty work is a contrast of His ministry in other placed where people were healed and reconciled to God. This didn't mean that Christ's power diminished in the presence of the people, but that His power could not affect them because they refused to believe in Him as the Son of God.

Using my comment earlier, the people during that time were actually blessed to physically experience Jesus yet they still remained hard-hearted towards Him. This shows that no matter how powerful God is if people want nothing to do with Him then they won't experience him.

It's like aspirin. Sure it's powerful AND effective, but unless someone with a headache or body pain takes it, it won't work.

The incident in Mark 6:1-6 also appears in Matthew 13:53-58 which records Jesus’ words to the disbelieving Jews in the synagogue:

"Only in his home town and in his own house is a prophet without honour." And he did not do many miracles there because of their lack of faith" (Matthew 13:57-58).

The answer to your question is to be found in the NIV Study Bible notes on Mark 6:5:

It was not that Jesus did not have power to perform miracles at Nazareth, but that he chose not to in such a climate of unbelief.

You ask how does Jesus addressing the people’s unbelief work. Fast forward to Mark 9:24, after the transfiguration of Jesus. A man asked Jesus to heal his son after Jesus’ disciples were unable to do so. Jesus responded initially by saying “O unbelieving generation, how long shall I stay with you? How long shall I put up with you? Bring the boy to me.” The boy’s father said to Jesus, “If you can do anything, take pity on us and help us.”

"IF you can?" said Jesus. "Everything is possible for him who believes." Immediately the boy’s father exclaimed, "I do believe; help me overcome my unbelief!" (Mark 9:23-24)

It has nothing with Jesus’ inability to perform mighty works (which he did, including raising the dead). It is about Jesus withholding miraculous power from those who lacked faith in Him.

If a person becomes a star athlete or becomes rich and famous and brings wealth, prosperity or esteem to their hometown the town usually calls them a "son". Jesus, the carpenters boy, was going around saying controversial things and attributing important prophesy to himself. Jews are raised to always anticipate and pray for the coming of their messiah.

When Jesus appeared they basically said, "...this is it??". His first miracle was turning the towns supply of about 30 gallons of water earmarked for their important ritual cleansing mitzvah baths into wine. The baths were more important to the women than the men and he performed the miracle at a wedding.

Their hometown son did controversial things. What I think many overlook is the ending of the passage that says since he could do very little miracles he went through the villages teaching. Being TAUGHT by Jesus himself had to be the most glorious and enlightening of experiences. He didn't give them a fish for the day, he tought them how to fish instead. What people should wonder is how many from his hometown ended up in the upper room at Pentecost or eventually became true believers after his lessons finally began to take root.

Jesus did nothing except the Father showed Him. The works He did were the works of the Father. Likewise the things He said. Wherever He went and whatever He did or said was given to Him from the Father, led and and enabled by the Spirit.

Joh 5:17  But Jesus answered them, My Father worketh hitherto, and I work. 

Joh 9:4  I must work the works of him that sent me, while it is day: the night cometh, when no man can work. 

Joh 14:10  Believest thou not that I am in the Father, and the Father in me? the words that I speak unto you I speak not of myself: but the Father that dwelleth in me, he doeth the works. 

Joh 5:19  Then answered Jesus and said unto them, Verily, verily, I say unto you, The Son can do nothing of himself, but what he seeth the Father do: for what things soever he doeth, these also doeth the Son likewise.  Joh 5:20  For the Father loveth the Son, and sheweth him all things that himself doeth: and he will shew him greater works than these, that ye may marvel. 

Joh 5:30  I can of mine own self do nothing: as I hear, I judge: and my judgment is just; because I seek not mine own will, but the will of the Father which hath sent me. 

Joh 8:28  Then said Jesus unto them, When ye have lifted up the Son of man, then shall ye know that I am he, and that I do nothing of myself; but as my Father hath taught me, I speak these things. 

Joh 12:49  For I have not spoken of myself; but the Father which sent me, he gave me a commandment, what I should say, and what I should speak. 

Christ had the Spirit without measure and therefore had perfect communion with the Father. It follows therefore that in all things He was led of the Spirit, the one possible exception (not really an exception but a heightened or strngthened example) being when He was driven of the Spirit to the temptations ~ Mar 1:12  And immediately the Spirit driveth him into the wilderness.  Mar 1:13  And he was there in the wilderness forty days, tempted of Satan; and was with the wild beasts; and the angels ministered unto him. 

We are told that at His baptism the Spirit descended upon Him and lighted upon Him and abode with Him and in particular that, Luk 4:1  And Jesus being full of the Holy Ghost returned from Jordan, and was led by the Spirit into the wilderness,  Luk 4:2  Being forty days tempted of the devil. And in those days he did eat nothing: and when they were ended, he afterward hungered. 

This is the very first thing after his baptism and sets the complete pattern for all things after.

So we see the Father had everything planned out and arranged everything and Jesus simply walked in the will of the Father and worked the good works, 'which God hath before ordained that He should walk in them.' You notice the extraction from Ephesians 2? Eph 2:10  For we are his workmanship, created in Christ Jesus unto good works, which God hath before ordained that we should walk in them. 

So then we too are to learn to walk as Christ walked, in entire and utter dependence upon the Father, 1Jn 2:6  He that saith he abideth in him ought himself also so to walk, even as he walked. 

After all without Him we can do nothing, absolutely nothing, zilch. Joh 15:5  I am the vine, ye are the branches: He that abideth in me, and I in him, the same bringeth forth much fruit: for without me ye can do nothing. 

Because we do not have perfect communion but are hindered by the old flesh life then it is necessary that we deny self and take up our cross and follow that our old flesh life be cut off and we live by the Spirit.

As it was for Christ so it is to be for us.

So the power to perform miracles, as you put it, was entirely with the Father and ordered of the Father. Christ limited Himself to live only as a man to fully take upon Himself our condition and show it was possible to live like that. It is a mistake, sadly all too common, to see that power residing in Him to use as He might choose, and that is an example to us. This is a very important point, which also addresses the false assumption in your question. The Father's will is perfect, Christ submitted Himself totally to that, why then should any not also want that?

Thy kingdom come. Thy will be done in earth, as it is in heaven.

As to why the Father did no miracles with some, the Father sees the hearts and since all miracles glorify God, whether those hearts are right to return glory to Him. There is an important principle here also. God never forces any, but will affirm those who seek Him and wait upon Him. A miracle to one not seeking or waiting upon Him would have the effect of forcing them. Likewise Jesus never openly asserted Who He was but always confirmed to those who had received revelation from the Father as to Who He was.

So the Father saw the hearts and would not work miracles for those who would be forced by them.

Remember those forced against their will remain of the same heart and it is a new heart that is required.

This also is why Jesus spoke in parables so as to leave it open for people to draw what they will or not from them but that those with a right heart would have understanding from the Father. Mat 13:13  Therefore speak I to them in parables: because they seeing see not; and hearing they hear not, neither do they understand.  Mat 13:14  And in them is fulfilled the prophecy of Esaias, which saith, By hearing ye shall hear, and shall not understand; and seeing ye shall see, and shall not perceive:  Mat 13:15  For this people's heart is waxed gross, and their ears are dull of hearing, and their eyes they have closed; lest at any time they should see with their eyes, and hear with their ears, and should understand with their heart, and should be converted, and I should heal them. 

So a contrite heart is vital.

Psa 34:18  The LORD is nigh unto them that are of a broken heart; and saveth such as be of a contrite spirit.

Psa 51:17  The sacrifices of God are a broken spirit: a broken and a contrite heart, O God, thou wilt not despise.

Isa 57:15  For thus saith the high and lofty One that inhabiteth eternity, whose name is Holy; I dwell in the high and holy place, with him also that is of a contrite and humble spirit, to revive the spirit of the humble, and to revive the heart of the contrite ones.