ATLANTA — Should Maria Palacios, a Mexican-born Georgia resident who moved to the state as a child and became an American citizen in June 2017, be allowed to run for the State House of Representatives?

She has been fighting for that right since she was thrown off the ballot in May, first by an administrative law judge, then by the Georgia secretary of state, Brian Kemp. Her legal battle comes against the backdrop of a state whose population is rapidly changing, a governor’s race with illegal immigration as a white-hot issue among conservatives, and a nationwide fear among liberals that their opponents are rigging the system to keep people of color out of the voting booths — and out of office.

Ms. Palacios, 28, hit another roadblock in an Atlanta courtroom on Wednesday, when a Superior Court judge agreed with Mr. Kemp that she was ineligible to appear on the November ballot as the Democratic candidate in the 29th District.

Mr. Kemp is running for governor, and facing a runoff on July 24 for the Republican nomination. He has attracted national attention — and a boost in his poll numbers — with ads that showed him brandishing a shotgun and boasting that he might use his pickup truck to round up “criminal” illegal immigrants and “take ’em home myself.”