No. New York City does not have a separate governor.
Unlike U.S. states (which have governors) or some federal districts (e.g., D.C. has a Mayor with governor-like powers), NYC is a city, not a state. Its chief executive is the Mayor. The State Governor oversees state-level functions affecting NYC: transportation (MTA), education policy, courts, National Guard, etc.
NYC was the U.S. capital (1785–1790) and New York State's capital until 1797 (moved to Albany). It has never been a separate sovereign entity with a governor.