N.Y. Second Class Cities Law § 51
Whenever there shall be a vacancy in the office of mayor, or whenever by reason of sickness or absence from the city the mayor shall be prevented from attending to the duties of the office, the president of the common council shall act as mayor and possess all the rights of mayor during such period of disability or absence. In case of a vacancy in the office of mayor he shall so act until noon of the first day of January next succeeding the election at which the mayor's successor shall be chosen. It shall not be lawful for the president of the common council when acting as mayor in consequence of the absence or sickness of the mayor to exercise any power of appointment or removal from office unless such sickness or absence shall have continued for a period of thirty days; or to sign, approve or disapprove any ordinance or resolution unless such sickness or absence shall have continued for a period of at least nine days.
N.Y. Second Class Cities Law § 51