New York Consolidated Laws, Second Class Cities Law - SCC § 15. Vacancies

If a vacancy shall occur, otherwise than by expiration of term, in an elective office of the city, including that of supervisor, the mayor shall appoint a person to fill such vacancy.  The person so appointed to such vacancy shall hold office until the commencement of the political year next succeeding the first annual election after the happening of the vacancy at which a successor can be elected for the balance of the unexpired term, if any.  If a vacancy shall occur in an appointive office of the city, otherwise than by expiration of term, the officer, officers, board or body authorized to make appointment to such office for the full term shall appoint a person to fill such vacancy for the balance of the unexpired term.