These voters, several of whom expressed support for abortion rights but said they would still vote for an anti-abortion congressman, revealed the uphill battle Ms. Cisneros faces as she tries to convince voters to oust a familiar political figure whose family has long been a fixture in the community. Even among Democrats, support for abortion rights may not uniformly motivate voters, particularly working-class Latinos — a demographic that has shown signs of drifting away from the party.

Hector Gomez, 67, has known Mr. Cuellar since they were classmates at J.W. Nixon High School in the 1970s, and has voted for the congressman in every election since he first ran in 2006.

“He’s doing his job,” said Mr. Gomez, an antique store owner, adding that although he is Catholic and opposed to abortion, the issue does not determine his vote. “Mr. Cuellar is the best choice because he’s not someone you can just brush off.”

Texas’ 28th Congressional District stretches from the Mexican border to San Antonio, and Laredo is its political center. A working-class city, it has been a Democratic stronghold for decades but remains culturally conservative, with residents who fill Catholic church pews on Sundays. Many describe themselves as apolitical, and said they are more focused on making ends meet than staking out positions on partisan political issues.

Before this week’s leaked Supreme Court opinion, abortion had not been the central issue in the primary campaign, though several national abortion rights groups had invested heavily in the district, focusing on the new state abortion restrictions in Texas. Ms. Cisneros did not run a television advertisement on the issue until late last month, according to AdImpact, an ad tracking firm. Until the focus on abortion was renewed this week, the runoff had been a mostly sleepy affair, with observers predicting an extremely low turnout.

Now, Ms. Cisneros and her supporters have moved to use the threat to abortion rights as a primary motivator for both voters and donors.

“We’re really at a moment where people are fired up and they know how much they are at risk of losing,” said Kristin Ford, the vice president of communications and research at the abortion rights group NARAL Pro-Choice America, which has sent organizers to Laredo to campaign for Ms. Cisneros.