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Hillary Rodham Clinton acknowledged on Tuesday that her use of a private email server while she was secretary of state was a “mistake,” and apologized directly for it, uttering words that many of her allies have waited to hear from her in hopes that she can quell a controversy that has dogged her presidential candidacy for months.

“That was a mistake. I’m sorry about that. I take responsibility,” Mrs. Clinton said in an interview with David Muir of ABC News broadcast Tuesday night. “And I’m trying to be as transparent as I possibly can.”

Asked by Mr. Muir about a recent poll in which respondents used words like “liar” and “untrustworthy” to describe her, Mrs. Clinton conceded that she still had work to do: “Obviously, David, I don’t like hearing that,” she said. “I am confident by the end of this campaign, people will know they can trust me, and that I will be on their side and I will fight for them and their families. But I do think I could have and should have done a better job answering questions earlier. I really didn’t, perhaps, appreciate the need to do that.”

When asked if she had ever second-guessed her decision to make another run at the White House, Mrs. Clinton began to choke up, admitting that she had, at times, before invoking her mother’s admonitions to “fight for what you believe in, no matter how hard it is.”

“I think about her a lot. I miss her a lot. I wish she were here with me,” Mrs. Clinton said of her mother, who died in 2011. But, she added: “I don’t want to just fight for me. I mean, I could have a perfectly fine life not being president. I want to fight for all the people like my mother who need somebody in their corner. And they need a leader who cares about them again. So that’s what I’m going to try to do.”

Mrs. Clinton’s apology on ABC was the more striking for coming just a day after an interview with The Associated Press in which she maintained that she did not need to apologize for her private email account and server, saying, “What I did was allowed.”

And in an interview with Andrea Mitchell of NBC News on Friday, Mrs. Clinton, asked if she was sorry, allowed only that she was “sorry that this has been confusing to people and has raised a lot of questions.”

In an Aug. 26 news conference, Mrs. Clinton said she understood why people had questions about the email arrangement, which she said came about as a matter of convenience so she could carry a single mobile device. She said she took responsibility for the decision to use the private server and said it would have been better to have used a private email only for personal matters and an official one for work.

Last week, Mrs. Clinton’s aides showed a video of that news conference to a New Hampshire focus group of independents and Democrats, according to a Democrat briefed on the focus group whose account was confirmed by a person in her campaign. Participants said they wanted to hear more from Mrs. Clinton about the issue.

The focus group also showed that the email issue was drowning out nearly everything else that Mrs. Clinton was hoping to communicate to voters — something Mrs. Clinton and her husband have complained about to friends.

Privately, some of Mrs. Clinton’s allies have drawn comparisons between her resistance to using the word “mistake” over the email server and her similar reluctance to say she had erred in voting as a senator to support the invasion of Iraq. That vote dogged her in the 2008 presidential primary, but Mrs. Clinton resisted calling it a mistake, despite entreaties from many liberals and some of her own aides.

Only in her 2014 memoir, “Hard Choices,” did Mrs. Clinton say she had “got it wrong” on the Iraq invasion.

“In our political culture, saying you made a mistake is often taken as weakness when in fact it can be a sign of strength and growth for people and nations,” Mrs. Clinton wrote.

“I thought I had acted in good faith and made the best decision I could with the information I had. And I wasn’t alone in getting it wrong,” she wrote. “But I still got it wrong. Plain and simple.”

In recent weeks, some advisers had privately expressed hope that Mrs. Clinton would acknowledge a mistake on her email practices in similarly clear and blunt terms.

But others on her team, saying they were bound by the constraints of a complex situation, with several investigations underway, argued that Mrs. Clinton was limited in her ability to defend herself.

For her part, Mrs. Clinton had long insisted that the controversy was a news media fixation that voters had not raised with her on the campaign trail. At a Democratic dinner in Iowa last month, she even made light of it, with a joke about the iPhone application Snapchat, whose pictures delete themselves. And a few days later, when a Fox News reporter asked if she had wiped her server of data, Mrs. Clinton quipped, “What, like with a cloth or something?”

Mrs. Clinton, who has said that she broke no rules and is trying to be as transparent as possible, turned over about 55,000 pages of emails to the State Department, which is reviewing them to comply with freedom-of-information lawsuits. She has said she deleted the remaining 31,000, which she deemed not work-related.

But the email controversy has stayed in the headlines, with new reports about whether Mrs. Clinton’s server contained classified information, the F.B.I.’s taking possession of the server to ensure its security, and the decision by the technician who maintained the server for Mrs. Clinton to invoke his Fifth Amendment rights against self-incrimination. (She says no material she sent or received was marked classified at the time.)

In recent days, Mrs. Clinton’s aides have signaled that she planned to address the email controversy more openly, and with a tone of humility rather than defensiveness.

Hillary Clinton to Show More Humor and Heart, Aides Say Staff members say the noise and distractions of modern campaigning have obscured these sides of Mrs. Clinton’s personality.