"There is definitely a competition on for the second place finish," said Mark Baldassare, president of the PPIC.
The poll finds that Cox, a San Diego businessman, has solidified the support of Republicans, who favor him by 22 points over Orange County Assemblyman Travis Allen.
Allen is followed by Democrats John Chiang and Delaine Eastin in the poll.
Cox was endorsed by President Donald Trump on May 18, a move Cox hopes will rally the state's Republican voters behind him.
Baldassare said the announcement didn't cause a major jump in the polling, which ran from May 11 until May 20, but could provide a decisive boost for Cox in the campaign's stretch run.
"I would be watching Villaraigosa's use of television commercials," Baldassare added. "And to what extent John Cox's endorsement by President Trump plays a role in these next couple of weeks."
Cold Water on California's Blue Wave?
The PPIC survey raised a red flag for Democrats running in California's 10 house seats labeled 'competitive' by the nonpartisan Cook Political Report.
Sixty-one percent of likely voters in these districts said they would support or 'lean more toward' supporting a generic Republican candidate over a Democrat.
"Democrats have their work cut out for them in these mostly Republican districts," Baldassare said. "I think these results just underscore the work ahead."
The poll includes prime Democratic pickup opportunities in Orange County districts where Republican incumbents have retired, but also districts in the Sierra foothills and East San Diego county, which have historically favored Republicans by wide margins.
Likely voters in the 10 districts told the PPIC that they'd prefer candidates who "work with the Trump administration" by a 2 to 1 margin over candidates who "push back" against the White House.