The phrase "primary opponent" refers to a person or candidate who is running against someone within the same political party in an election to become the party's representative in the final election. Full definition
Neither of his two Democratic primary opponents participated in our screening panel. (the504dems.org)
This sharing comes, of course, from the candidate who transformed a Republican primary opponent into a furry demon - sheep in a television ad. (thecaucus.blogs.nytimes.com)
These Republican candidates, several of whom are also trying to primary their opponents for the Democratic line, hold no allegiance to Republican philosophies or values. (hudsonvalleyone.com)