While they are duking it out with the city at the policy level, the public school unions are also hoping to kick it up on the political front, as part of a new coalition, New Yorkers for Great Public Schools, that plans to weigh in on the 2013 mayoral race. The group was formed to counter StudentsFirstNY, which has said it will inject $10 million a year, raised from hedge fund managers and venture capitalists, into the race to press for more policies like those adopted under Mayor Michael R. Bloomberg.