Yes, there is a constitutional process.
The 150 members of the State Assembly, currently 106 of whom are Democrats, can impeach on a simple-majority vote. If they try and fail, I’d hate to be them. If they succeed, Cuomo is sort of “suspended” and the Lt. Gov. takes over, at least temporarily.
A trial would be held in an “impeachment court,” largely the same as the State Senate (with the addition of certain senior judges). In total there would be sixty-nine jurors; two thirds would have to vote for conviction. Currently more than two thirds can be considered Democrats. If the Senate convicts (or if Cuomo resigns), the Lt. Gov. takes over for the remainder of his term; and he can also be barred from seeking office again.
Whereas if he is acquitted, he continues in office, the Lt. Gov. goes back to being the Lt. Gov., and Albany will become totally uninhabitable.