"To run a primary" means to organize and hold an election within a political party to choose the party's candidate for an upcoming election. Full definition
The analysis compared the performance of test results of academies with local authority run primary schools in similar circumstances. (educationbusinessuk.net)
However, he said he would still consider running a primary if he is not endorsed because he believes he is the best candidate. (nystateofpolitics.com)
The cost of running primaries also seems to be a step in the wrong direction. (dprvoting.org)