The accusation that Ted Cruz used dirty tricks to win Iowa’s Republican caucuses has hounded him in New Hampshire and threatens to keep after him into South Carolina, inflicting damage on his campaign in that key Southern primary where he is a top contender.
Republican presidential front-runner Donald Trump has kept the story alive, jabbing Mr. Cruz when the Texas senator boasted of his caucus victory during the GOP candidate’s debate Saturday.
“That’s because you got Ben Carson’s votes, by the way, but we won’t say that,” said Mr. Trump, provoking a combination of hoots and cheers from the debate audience at Saint Anselm College in Goffstown, New Hampshire.
The candidate expected to gain the most from voters who sour on Mr. Cruz because of the allegation isn’t Mr. Carson or Mr. Trump — it’s Sen. Marco Rubio of Florida, according to political experts.
Mr. Rubio gained momentum after finishing third in Iowa, close behind Mr. Cruz and Mr. Trump. He is running second in New Hampshire and third in South Carolina, according to recent polls.
Inheriting support from Mr. Carson, who draws about 8 percent in South Carolina surveys, would be a major coup for Mr. Rubio.
The charge that the Cruz campaign siphoned off Mr. Carson’s supporters on caucus night last week by spreading a rumor that the retired neurosurgeon was preparing to quit the race has kept Mr. Cruz on a defensive footing.
Mr. Cruz insisted that his campaign merely relayed a CNN report about Mr. Carson taking a break from the campaign trail to return to his home in Florida, though he apologized to Mr. Carson for not quickly correcting speculation that he was poised to drop out of the race.
Mr. Carson called it a “dirty trick” that Cruz campaign workers used the story to urge his supporters to back Mr. Cruz rather than waste their vote. He demanded Mr. Cruz fire the campaign staff responsible, and Mr. Cruz refused.
“Washington ethics basically says if it’s legal, you do what you need to do in order to win. That’s not my ethics. My ethics is you do what’s right,” Mr. Carson said during the debate.
Turning to face his rival on stage, Mr. Cruz again apologized for the mix-up.
“When this transpired, I apologized to him then, and I do so now. Ben, I’m sorry,” he said.
The issue dogged Mr. Cruz in New Hampshire, where some voters compare it to the Watergate scandal that brought down President Nixon.
“It is good that he apologized, but I think that he is going to have to dismiss the workers that are involved there,” Phil Richardson, 67, said at a Cruz rally in Portsmouth. “We lived through the Watergate era, so we know what dirty tricks unpunished can lead to.”
At a town hall meeting in Nashua, New Hampshire, a voter pressed Mr. Cruz to clear the air about what happened.
“Behind you is a sign that says ‘TrusTed,’” the man said to the candidate. “You are known as a man with integrity. A lot of people are voting for you for your integrity. Tell us exactly what happened in Iowa with the Ben Carson situation, because I am getting all sorts of stuff on the news, and it’s very confusing.”
Mr. Cruz was never positioned as a leading candidate in the Granite State. He instead staked his campaign on the preceding and following contests in Iowa and South Carolina, the latter state holding its first-in-the-South primary Feb. 20, which historically distills lasting front-runners and propels them toward March’s delegate-rich multistate primaries.
The latest polls show Mr. Cruz in fourth place in New Hampshire but running second behind Mr. Trump in South Carolina, where he has a formidable organization and a team with strong ties to the conservative, evangelical-heavy state.
And South Carolina is where the dirty tricks charge hurts him most.
“It’s done a tremendous amount of damage,” said Joe Dugan, founder and executive producer of the South Carolina Tea Party Coalition.
He said the episode has hardened the rift between conservatives backing Mr. Cruz and conservatives backing Mr. Trump, as well as poisoned Mr. Carson’s evangelical supporters against Mr. Cruz.
Those Carson supporters will become pivotal if Mr. Carson’s flagging campaign ends after New Hampshire, said Bruce Haynes, a Republican strategist with expertise in South Carolina politics.
“How badly does that hurt Cruz’s ability to try to bring Carson voters into the fold? It could be really decisive,” he said.
He added: “The interesting thing there is that the favor that Trump is doing by driving that narrative is for Rubio and not for himself.”
“Cruz can pretty much give up on those guys,” said Moye Graham, the Republican Party chairman for South Carolina’s 6th Congressional District.
He said the issue has come up at recent Republican Party events in the state.
“It certainly has upset the Carson people in South Carolina, and it certainly has put the Cruz people on the defense,” he said.
• Seth McLaughlin in Hampton, New Hampshire, contributed to this report.