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St. Thomas Aquinas's commentary on Matthew 27:46 explains that
He calls Him His Father, in that He is God; He calls Him His God, in that He is man: wherefore, when He says, My God, My God, etc., it is clear that He is speaking insofar as He is a man; hence, He groans, to express the greatness of His human suffering.
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There are two thought as to what this Scripture implicates;
Jesus could have been crying out in pain as he accepted the sins of the World, and the accompanying separation from God the Father. By accepting the sins of the World Jesus made himself abhorrent to the Father, and as an unrepentant abhorrent to the Holy Ghost.
A lesser accepted idea is that Jesus, who often quoted Scripture, Was quoting:
Psalm 22:1 To the chief Musician upon Aijeleth Shahar, A Psalm of David. My God, my God, why hast thou forsaken me? why art thou so far from helping me, and from the words of my roaring?
There is no reason for ascribing either of ideas to Jesus at the Cross; as Jesus is one part of the Trinity and Eternal. Some mysteries of the Bible will never be solved on Earth.
I offer this answer only as what I have heard, but not as a real answer since the question is very close to being a truth question.