Ezekiel gave a prophecy that David would return.

And I will set up one shepherd over them, and he shall feed them, even my servant David; he shall feed them, and he shall be their shepherd. And I the LORD will be their God, and my servant David a prince among them; I the LORD have spoken it. Ezekiel 34:23-24 KJV

The book of Ezekiel was written centuries after David, the second king of Israel, had died. So was Ezekiel prophesying that David would return? If so, that would require his reincarnation.

Jesus is called the son of David, but is David’s Lord and not David.

And Jesus answered and said, while he taught in the temple, How say the scribes that Christ is the Son of David? For David himself said by the Holy Ghost, The LORD said to my Lord, Sit thou on my right hand, till I make thine enemies thy footstool. David therefore himself calleth him Lord; and whence is he then his son? And the common people heard him gladly. Mark 12:35-37 KJV

David is not the Lord, shown clearly here. According to the Lord Jesus, David called the Son of David his Lord. David and the Son of David are two different beings. David called the Son of David his Lord.

It would not be correct to call Jesus a reincarnated David. Instead, Jesus is the reincarnated Jehovah (the LORD in a human body).

The prophecy from Ezekiel says the LORD will be Israel's God, and David will be a prince among them, which implies the reincarnation of David.