Some Republicans lining up behind Mr. Walker questioned the latest allegations. Some didn’t seem to care either way.

“If y’all find a perfect candidate that has never had challenges in their life, I want you to bring them to me and let me meet him or her,” said Dominic LaRiccia, a Republican state representative backing Mr. Walker.

Marci McCarthy, the Republican chair of DeKalb County in the metro Atlanta area, said, “Ultimately Georgia’s voters will put their own lives and livelihoods first because, in either case, they are not voting for fathers and husbands of the year.”

Notably, Gov. Brian Kemp of Georgia, a Republican who has been leading his race for re-election, has kept his distance from Mr. Walker. The two have yet to hold a campaign event together this year. Asked about the episode on Tuesday, a spokesman for Mr. Kemp, Cody Hall, did not mention Mr. Walker and said only that the governor was “laser focused” on fund-raising and voter turnout in the final weeks of the campaign.

Mr. Walker has been found to have embellished or misrepresented key elements of his résumé, including claiming he worked in law enforcement, although he did not. For years, his food-distribution company said it would donate a portion of its earnings to charity, but there is little evidence that it did so. His convoluted and confusing stump speeches have made headlines for months.

“You’re not going to convince people who made it this far that Herschel’s really a bad guy,” said Randy Evans, a former Republican National Committee member from Georgia, who supports Mr. Walker and spent Tuesday morning at a Republican county breakfast in northwestern Georgia. “Today’s reaction was everybody offering their ideas how to respond to Warnock’s lies,” he added. “The electorate is hardened — on both sides.”

The base of the Republican Party, in particular, is primed to ignore allegations that appear in the mainstream media, especially anonymous ones, Republican officials say.

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