In many passages of Scripture, Jesus is taught to have a God (Psalm 45:6-7, Micah 5:2-4, Matthew 27:46, Mark 15:34, John 20:17, Romans 15:6: 2 Corinthians 1:3, 11:31, Ephesians 1:3, 1:17: 1 Peter 1:3, Hebrews 1:8-9, Revelation 1:5-6, 3:2, 3:12, and potentially Revelation 5:10).
Perhaps the most striking of these is Revelation 3:12, in which Jesus speaks of "my God" four times in quick succession:
Revelation 3:12 (NRSVUE)
12 If you conquer, I will make you a pillar in the temple of my God; you will never go out of it. I will write on you the name of my God and the name of the city of my God, the new Jerusalem that comes down from my God out of heaven, and my own new name.
So, what does having a "God" mean? When someone says, "my God" in this way, what do they mean? In every single instance in Scripture, someone speaking in this way is referring to the God that they worship - the God who is over them, and to whom they owe obeisance. And, Scripture teaches that there is one God, whose name is Yahweh. Therefore, by confessing that He has a God, who He always and everywhere speaks of as a separate subject from Himself, Jesus indicated that He Himself was not God.
Consider that the concept of God having a God does not make sense. There is a single God, who is the source of all else. He has none over Him. He does not worship a God, and prays to no one. He cannot call any "my God", because He is "the only true God" (John 17:3).
A very popular response to this point from those who believe in the doctrine of the Trinity is to ask, "If God were to become incarnate, would He be an Atheist?". In reply, no, God would not be an Atheist, because an Atheist is someone who does not believe in the existence of God, which He would know to be a false belief, if He incarnated. However, Jesus is not simply "not an Atheist". Rather, He speaks of His "God", prays to God, and everywhere speaks of God as a separate subject from Himself. If God were to incarnate, this would be highly unexpected. One would expect Him to say, "I am God, there is none else". One would expect Him to say, "I do not need to pray to anyone, I am God". It is highly unexpected that God, if He were to incarnate, would speak and be spoken of as having a "God", never once clarifying that He is that very same Being which He is calling His "God".
In summary, not only do the New Testament authors always and everywhere distinguish God from Jesus, but Jesus is also spoken of as Himself having a God, which teaches with even greater clarity that He is a distinct subject from God, and is not God.