Frank Curnow, a retired Navy veteran, had a question for the Republican Party volunteers at an early voting site in Daytona Beach, Fla., on Thursday.

Why, he asked, were some polls suggesting a tight race for the open seat in his reliably conservative congressional district, Florida’s Sixth?

“That’s not good,” said Mr. Curnow, 77, a registered Republican who voted for the party’s candidate, State Senator Randy Fine, because “he carries the Trump banner.” The race, he added, “is critical.”

Voters in the district, which includes the stretch of coast between St. Augustine and Daytona Beach, overwhelmingly voted for President Trump. But on Tuesday, they will cast ballots in what seems to be an unusually competitive contest to replace former Representative Michael Waltz, who stepped down to become Mr. Trump’s national security adviser.

Mr. Fine, who is facing Josh Weil, a Democrat and public school employee, is still widely favored to win. But with Republicans clinging to their House majority by just a few seats, it was clear this week that voters in the district are thinking about what the outcome of the race — and of another race for former Representative Matt Gaetz’s seat in the Florida Panhandle — could mean for Mr. Trump’s agenda.

“It’s important that we keep the seats,” said Doreen Colby, 74, a registered Republican who lives in Daytona Beach. “We want to make sure Trump can keep doing what he’s doing.”