Gov. Andrew M. Cuomo is doing his best to appear invincible as he seeks a second term. He generally avoids even mentioning that he faces a challenge in the Democratic primary, and with barely a month to go, he is making few public appearances.

But despite the governor’s above-the-fray image, his campaign is working outside of the spotlight on one urgent mission: trying to knock one of Mr. Cuomo’s long-shot challengers, a law professor named Zephyr Teachout, off the Sept. 9 ballot.

Mr. Cuomo’s campaign argues that Ms. Teachout has not lived in New York State long enough to be eligible. But Ms. Teachout says the challenge to her residency, which will be heard on Thursday in State Supreme Court in Brooklyn, is just an attempt to intimidate her, as well as to sap her campaign of time and resources.

“He wants me to just go away, because there is not supposed to be a primary in Andrew Cuomo’s New York,” Ms. Teachout, 42, said in an interview on Wednesday afternoon, which she spent in a Midtown Manhattan law firm, poring over piles of mail, bank statements and tax returns meant to document that she is a bona fide New Yorker.