Under what circumstances can the governor of New York be impeached?
There is no list of impeachable offenses, as such, in the New York State Constitution, or its state laws. Article 24 of the state constitution grants the assembly "the power of impeachment by a vote of a majority of all the members elected thereto", but does not go into specifics about what in particular consitutes an impeachable offense.
The closest such definition comes in Article 7-C, section 240 of New York's Judiciary Law:
§ 240. Its jurisdiction. The court for the trial of impeachments has power to try impeachments, when presented by the assembly, of all civil officers of the state, except justices of the peace, justices of justices' courts, police justices, and their clerks, for willful and corrupt misconduct in office.
Whether, for example, the allegations against Governor Cuomo fall under this definition will be for the court for the trial of impeachments to decide, if the New York Assembly approves articles of impeachment against him.
This is probably one of the least important questions in the whole process. Even when the law includes a definition of an impeachable offense (as the US constitution does), impeachment is inevitably highly political.
In this case, the state constitution has nothing to say about it. The article on impeachment is focused only on the procedure and case law is apparently very sparse, leading commentators to consider that “In New York, an impeachable offense is whatever the Legislature wants it to be.”