ATLANTA — An anti-Trump Republican advocacy group recently organized a focus group of G.O.P voters in Georgia to get their take on perhaps the most competitive and consequential primary election in the state. They heard a lot of indecision.

Most of the voters, convened by the group, the Republican Accountability Project, knew little about the race between Brad Raffensperger, Georgia’s secretary of state, and his leading challenger, Representative Jody Hice. Mr. Raffensperger seemed to get the benefit of the doubt — until the voters were reminded of the back story.

As the state official responsible for certifying the 2020 presidential election results, Mr. Raffensperger rejected President Donald J. Trump’s attempt to overturn his defeat. Mr. Trump recruited Mr. Hice to seek revenge.

“Go Jody, I guess?” said one voter.

Three out of six others agreed.

The exchange offered a glimpse into why the Republican primary race for the office that oversees elections remains a dogfight just days from Election Day, on May 24. Two years after Mr. Trump lost Georgia by the slimmest of margins and Democrats captured both of the state’s open Senate seats, wounds from the 2020 election have still not completely healed for some partisans.

But marshaling that residual anger to unseat an incumbent is not an easy feat. Mr. Raffensperger has worked to win back Republicans by casting himself as a defender of “election integrity,” even as he has spent hours debunking a laundry list of false claims about the 2020 election. Some voters’ memories and passions have faded. Many never had strong opinions about their secretary of state.

It all has made the race one of the purest tests yet of whether the 2020 election lie can be weaponized to win elections. While polls have shown that leagues of Republican voters in Georgia and elsewhere largely embraced the fiction that the 2020 election was “stolen” in its immediate aftermath, it is not clear those concerns alone, or Mr. Trump’s personal vendetta, are enough to drive voters’ choices.

“I think 2020 was really a turning point in how closely people looked at things,” said Salleigh Grubbs, chairwoman of the Cobb County Republican Party. “Before, people might not have even realized that the secretary of state was in charge of running the elections for the state. But now they’re keenly aware of it.”

Mr. Hice is one of more than a dozen candidates running for secretary of state under the America First banner, alongside others in battleground states like Arizona, Michigan and Ohio. They share an unflinching loyalty to Mr. Trump and a belief that the 2020 election was marred. Some are calling for a law enforcement arm to more aggressively prosecute violators of election laws.

Polls show large numbers of undecided voters in the race, with Mr. Hice and Mr. Raffensperger neck and neck, each with about one-third of the vote. Both campaigns are braced for a runoff.

Mr. Trump’s attempt at payback for 2020 in Georgia is floundering in the state’s other marquee primary on Tuesday. Former Senator David Perdue, his pick to challenge Gov. Brian Kemp after the governor refused to overturn the election results, is trailing Mr. Kemp in polls and fund-raising. Mr. Trump has hardly weighed in publicly on Mr. Perdue’s prospects since hosting a “tele-rally” for the former senator in April. His former vice president, Mike Pence, is set to visit the state to campaign for Mr. Kemp on the eve of the primary election.

Mr. Kemp has been adept at using his office to win over skeptical Republicans, passing a slew of conservative policies on elections, law enforcement and education. For voters still enthralled with false claims of fraud, Mr. Kemp can point to the Election Integrity Act of 2021, which limits provisions like ballot drop boxes and mobile voting centers.

“When voters see that kind of activity around the concern they have, it just becomes difficult to drive an argument that people who are in office are being inattentive to the issue,” said Brad Alexander, an Atlanta-based political consultant and Raffensperger supporter, who was among several who argued that the potency of the “stolen” election debate has started to wane.

In January, 43 percent of Georgia Republican voters said they were confident that the November elections would be fair and accurate, according to a University of Georgia poll. By April, that number had increased to nearly 60 percent. And a record number of voters have already participated in the state’s primary elections, topping more than 700,000 voters on the final day of early voting.

Mr. Hice, a four-term congressman, was one of the 147 House Republicans who voted against certifying the election results for President Joe Biden. He later took part in a White House meeting alongside Representative Marjorie Taylor Greene to try to determine how the election results could be flipped in Mr. Trump’s favor. He has claimed, falsely, that Mr. Trump would have won in Georgia if the election had been “fair.”

At a meeting with the Atlanta Young Republicans on Thursday, Mr. Hice made unsupported claims about “ballot harvesting,” said he no longer wanted to use Dominion Voting machines and slammed Mr. Raffensperger for sending out unsolicited absentee ballots ahead of the last election. The voice of the people had been “violated” in 2020, he said.

County officials in Georgia identified 64 cases of potential fraud out of the state’s roughly 5 million votes in the 2020 election, according to an Associated Press survey of all but 11 of the state’s 159 counties.

But Mr. Hice has tried to make his case about more than just 2020 — raising the prospect of fraud in the future.

“The issue is not who wins an election, but the issue is absolutely, Was it a fair election?” he said. “If the election itself is compromised or violated, then all the effort out there really doesn’t matter anymore. And that’s what we’ve got to defend. That’s what we’ve got to protect at all costs.”

Most of the young Republicans in the room said they wanted to hear what Mr. Hice would do differently from Mr. Raffensperger in elections should he be elected.

“Even if you’re not a dogmatic, election-was-stolen person, there are a lot of people with reasonable doubt, and that reasonable suspicion is fair,” said Chris Campbell, 39, a national accounts manager at SmartFeeds. “Raffensperger didn’t address those concerns well, and people don’t have confidence in him. But I do have confidence in Hice and trust that he would run the office with integrity.”

But there were clear divisions over that conclusion.

“Questions around the 2020 election — like, Was it stolen, or was it not stolen? — aren’t healthy,” said Allen English, the president of the Atlanta Young Republicans, who remained undecided between the candidates. “Looking back at 2020 isn’t productive.”

Many Republicans declared Mr. Raffensperger all but defeated as soon as he announced he would run for re-election, citing Mr. Trump’s ire and that of many voters. Mr. Raffensperger was censured at a state G.O.P. convention last year. But Mr. Trump has been unexpectedly quiet about the race, beyond lumping Mr. Raffensperger in with Mr. Kemp in statements railing against what he calls the R.I.N.O.s — Republicans in name only — running the state.

Sarah Longwell, the founder of the Republican Accountability Project, which conducted the focus group last week, said that Mr. Trump’s limited involvement could be playing a role in some voters’ indecision.

Mr. Raffensperger has raised more money than his competitors, courting far-right voters and touting what he calls “true conservative” values. At an event hosted by the Buckhead Young Republicans, Mr. Raffensperger discussed the 2020 election, but said his office was focused on “the real issue” of preventing noncitizens from voting. Mr. Raffensperger has called for amending the state Constitution to prohibit noncitizens from voting, something that state law already forbids. A state review found no instances of ballots being cast by noncitizens in 2020.

Mr. Raffensperger acknowledges that his pitch has done little to pacify some Republicans.

“I was never Public Enemy No. 1. I just think that some people weren’t pleased with the election results,” he said. “But in this race here, I’m leaving it to the goodness of my fellow Georgians who stood for the law, made sure that we followed the Constitution and follow the law. And that’s all I can do.”

The roughly 15 people present at the event — mostly 20-something conservatives — were receptive to his message. While some still had concerns about election oversight, they said they appreciated his candor in the face of ongoing attacks.

“There wasn’t anything that anyone else would have done to do anything,” said Bradley Schober, 27, an attorney who said he had already voted for Mr. Raffensperger. “Ultimately, the guy took as much heat as anybody and came out on the other side standing.”

Despite the high interest in the showdown, less than $1 million has been spent on television advertising in both the Democratic and Republican primary races for Georgia’s secretary of state, according to AdImpact, an ad-tracking firm, with the candidates exchanging attacks over the outcome of the 2020 presidential election and voting integrity.

Mr. Hice’s ads have showcased footage of him and Mr. Trump at a rally in Georgia, where the former president called Mr. Raffensperger a R.I.N.O. and “one of the worst secretary of states in America.”

Mr. Raffensperger’s ads have tried to rally Republicans against a common opponent — Stacey Abrams, the Democratic nominee for governor. The group she founded, Fair Fight, sued the state over discriminatory election practices in 2018. The lawsuit is currently in court.

“When Stacey Abrams attacked Georgia’s elections, Jody Hice did nothing,” the ad’s narrator says. The biggest spender on broadcast television has been Bee Nguyen, the leading candidate in the Democratic primary who has cast herself as a bulwark against Republican efforts to restrict voting access and meddle in elections.

“When our voting rights are attacked,” Ms. Nguyen says in one ad, “we all lose.”

Tariro Mzezewaand Neil Vigdor contributed reporting.