The Text

1839 Constitutional Amendment:
Mayors of the several cities in this State may be elected annually by the male inhabitants entitled to vote for members of the common council of such cities respectively, in such manner as the legislature shall by law provide; and the legislature may, from time to time, make such provision by law for the election of any one or more of such mayors; but until such provision be made by law, such mayor (excepting the mayor of the city of New-York) shall be appointed in the manner now prescribed by the Constitution of this State; and so much of the tenth section of article fourth of the Constitution of this State as is inconsistent with this amendment is hereby abrogated.

The full provision includes both the amendment and the text it amends – NY Constitution 1833, Article IV, Section 10:
The mayors of all the cities in this state shall be appointed annually, by the common councils of the respective cities. Except in the city of New-York, in which the mayor shall be chosen annually by the electors thereof qualified to vote for the other charter officers of the said city, and at the time of the election of such officers.