Is Jesus speaking from Himself or is the Holy Spirit speaking through Him?

Perhaps the difficulty with these biblical trinitarian conundrums stems from a lack of appreciation for the economical aspect of the Trinity and an overemphasis on the essential Trinity.

Augustine helps us here from "the Trinity":

Not that the voice could be produced without the activity of the Son and Holy Spirit (the triad works inseparably); but it was produced to manifest the person of the Father alone, just as the three produced that human being of the virgin Mary and yet it is the person of the Son alone—the invisible three producing what is the visible person of the Son alone. (110)

Hence, the three operate inseparably to manifest any one of the three distinctly, and in every action of one the three are understood to be operating simultaneously.

The reason for this is very simple: In His doing the Trinity is unitary, not triple. Thus, none of the three ever acts independently of the other two. Whatever one does, the other two also do with Him.

For example, in the incarnation the three of the Trinity act, not just the Son, as we might be tempted to think. Matt 1:35 shows the angel that appeared to Mary foretold the triune action of God in the incarnation:

The angel answered and said to her, The Holy Spirit will come upon you, and the power of the Most High [alluding to the Father] will overshadow you; therefore also the holy thing which is born will be called the Son of God [Son also implies the Father's begetting]

With regards to His speaking [the OP], we can see from His words to the Pharisees:

But if I, by the Spirit of God, cast out the demons, then the kingdom of God has come upon you (Matt. 12:28)

His work of manifesting God's kingdom on the earth (that is, not His own) was fully by the Spirit of God (that is, not by His own might and strength, even as the Son of God).

Furthermore, in the Gospel of John we find particular declarations by the Son regarding His coinherence with the Father in His work:

He who sent Me is with Me; He has not left Me alone, for I always do the things that are pleasing to Him” (8:29).

He who has seen Me has seen the Father; how is it that you say, Show us the Father? Do you not believe that I am in the Father and the Father is in Me? The words that I say to you I do not speak from Myself, but the Father who abides in Me does His works.” (John 14:9-10).

We should not think that in coming to dwell among men and as man, the Son departed from the Father and was hence separate from Him. Or the Spirit for that matter.

A careful examination of the Scriptures will indicate in instance after instance that the three of the trinity, while distinct in their respective hypostases, operate as one in their actions.

Thus, whatever one of the Trinity does must be understood as being done by the other two as well. There is never an action of one of the Trinity that is independent of the other two. While the personal distinctions among the three are maintained, any operation of the Trinity is one operation, and hence when one acts, the other two are identified with the one. Because of this, particularly in the economy of God the distinctions among the three are less acutely maintained in the biblical record.

Jesus spoke for the Father.

John 12:49 says,

"For I did not speak on My own initiative, but the Father Himself who sent Me has given Me a commandment as to what to say and what to speak."

John 14:10 says,

"Do you not believe that I am in the Father, and the Father is in Me? The words that I say to you I do not speak on My own initiative, but the Father abiding in Me does His works."

So the words that Jesus spoke were the Father's words. Now, how he came to know and speak those words is that the Spirit of the Father rested upon him. The words that he spoke are the Fathers words, and he only spoke the words that the Father bade him speak, and did the things that the Father bade him do.