Some public schools have some or all of their board members appointed by the mayor and/or governor, according to the Education Commission on the States:

Oakland, CA
3 of the 10 local school board members for the Oakland school district are appointed by the mayor, and 7 of the 10 members are elected.

Chicago, IL
Local school board members for the Chicago school district are appointed by the mayor of Chicago.

Indiana
While members of 274 local school boards are elected, members of 16 local school boards are appointed, depending on the school district, by the mayor, the county commissioner, the city council or a combination of these individuals and entities.

Baltimore, MD
School board members for the Baltimore city school district are jointly appointed by the governor and the mayor of Baltimore

Boston, MA
Local school committee members for the Boston school district are appointed by the mayor of Boston.

Detroit, MI
In the Detroit school district, 6 of the 7 local school board members are appointed by the mayor of Detroit, and 1 of the 7 members is appointed by the governor.

New York, NY
In New York City, 5 of the 13 members are appointed by the 5 borough presidents and 8 of the 13 members, including the chancellor of public instruction who serves as chairperson, are appointed by the mayor.

Yonkers, NY
Yonkers city school district board members are appointed by the mayor.

North Carolina
While members of 14 city school boards and 100 county school boards are elected, members of 3 city school boards are appointed by city councils.

Cleveland, OH
The members of the municipal school district board in the Cleveland school district are appointed by the mayor of Cleveland.

Philadelphia, PA
In Philadelphia, a 5-member school reform commission consists of 3 appointees of the governor and 2 appointees of the mayor.

Providence, RI
Local school board members for the Providence school district are appointed by the mayor

Source:
Local School Boards.  Education Commission of the States.  Denver, CO.  2003