Several Boston city councilors questioned the leadership of School Superintendent Carol R. Johnson on Tuesday, a day after she publicly apologized for failing to take action when one of her headmasters was arrested for assaulting his wife.

Johnson said she “made a mistake” by not placing Rodney ­Peterson on administrative leave after his June 17, 2011, arrest and sent the wrong message about ­domestic violence when she wrote a glowing letter on Peterson’s ­behalf to the judge who sentenced him to probation.