MIAMI — Coming off her stunning loss in Michigan, Democratic front-runner Hillary Clinton received the most pointed questions yet during this election cycle about her email server and the Benghazi terrorist attack.

Moderator Jorge Ramos of Univision asked at Wednesday night’s debate whether she would drop out of the race if an FBI investigation into the email system produces an indictment against her.

“For goodness sake, that is not gonna happen. I’m not even answering that question,” Mrs. Clinton said in a testy exchange early in the debate at Miami-Dade College.

Mr. Ramos also asked Mrs. Clinton why she sent an email to her daughter the night of the Benghazi attack saying it was a terrorist action, while explaining to the victims’ parents it was incited by a video. The question elicited boos from the audience.

“We were scrambling to get information that was changing literally by the hour,” Mrs. Clinton explained. “And when we had information, we made it public, but then sometimes we had to go back and say we had new information that contradicts it.”

Both Mrs. Clinton and Sen. Bernard Sanders, her rival for the Democratic presidential nomination, denounced Republican front-runner Donald Trump, although both dodged a question asking outright whether Mr. Trump is a racist.

Mrs. Clinton took a jab at his slogan: “Make America Great Again.”

“You don’t make America great by getting rid of everything that made America great,” Mrs. Clinton said.

Mr. Sanders said the public will never elect Mr. Trump to the White House, and said in every head-to-head poll with either him or Mrs. Clinton versus Mr. Trump, the Democrats win.

“I think that the American people are never going to elect a president who insults Mexicans, who insults Muslims, who insults women, who insults African-Americans,” Mr. Sanders said. “And let us not forget that several years ago, Trump was in the middle of the so-called birther movement, tried to delegitimize the president of the United States of America.”

Both candidates said they support a path to citizenship, with Mrs. Clinton saying she’d make it one of her priorities in her first 100 days if elected to the White House.

“I am staunchly in favor of comprehensive immigration reform and have been so in all of my public career,” Mrs. Clinton said. She also attacked Mr. Sanders for voting against comprehensive immigration reform in 2007, while she voted for it.

Mr. Sanders hit back, saying at the same time in 2007, Mrs. Clinton was against giving drivers licenses to undocumented workers, a position she’s since reversed herself on. He also defended his vote against immigration reform, citing proposed unfavorable conditions for guest workers in the bill.

Mr. Sanders also accused Mrs. Clinton of turning her back on illegal immigration children when a flood of them were entering the U.S., fleeing violence in Honduras.

“They came into this country, and I said welcome those children into this country. Secretary Clinton said, ‘Send them back,’ ” Mr. Sanders said.

Mrs. Clinton insisted that she wanted to protect the children from the dangerous trip to the U.S. and spare them from detention in the U.S.

Both Mr. Sanders and Mrs. Clinton accepted demands by Mr. Ramos for a pledge not to deport any children or anyone without a criminal record if they became president.

Mr. Sanders‘ pointed attacks on Mrs. Clinton demonstrated that he was unwilling to cede Hispanics, who have been a major part of the coalition supporting the former first lady.

The debate was their first after Mr. Sanders‘ shocking win in Michigan, which kicked off the discussion as Mrs. Clinton was asked what went wrong in Michigan. Mrs. Clinton deflected, saying her campaign is focused on the long run.

“It was a very close race. We won some, lost some, but I’m very pleased with the overall outcome yesterday,” Mrs. Clinton said from the stage. “I’m continuing to work hard for every single vote across our country. I was pleased I got 100,000 more votes last night than my opponent.”

She said her campaign was prepared for a marathon race, which plays in all states and is inclusive. Although Mrs. Clinton lost Michigan, she defeated Mr. Sanders handily in Mississippi.

Mrs. Clinton also handled a hard question when she was asked why so many people have the impression she is untrustworthy and dishonest, blaming her lack of “natural” political skills for leaving people with a bad impression that she is untrustworthy and dishonest.

She said that it was “painful” when she hears that a majority of Americans think that she can’t be trusted.

Mrs. Clinton said she “took responsibility” for her actions in public life that would cause people to doubt her, but she insisted that her intention has always been to help people.

“I also have very much committed to the best of my ability to helping people,” she said. “That is something I care deeply about.”

“I’m not a natural politician like my husband or like President Obama,” she said. “I hope people see that I am fighting for them.”

Mr. Sanders dealt Mrs. Clinton a surprise blow on Tuesday night after winning Michigan by 2 percentage points, even though he had trailed the former secretary of state by double digits in every poll taken before the contest.

Losing Michigan wasn’t a fatal blow to the Clinton campaign, although it does give Mr. Sanders the fuel he needs to continue competing until the convention. It also highlights the need for Mrs. Clinton to sharpen her economic messaging in the Rust Belt states to woo independent voters, white men and younger voters — among all of which groups she lost to Mr. Sanders in Michigan.

Six in 10 Michigan voters thought trade agreements like the North American Free Trade Agreement led to U.S. job losses, and of those voters, six in 10 voted for Mr. Sanders. Mrs. Clinton continues to be plagued by her past support for trade deals, which many voters in places like Illinois, Ohio and Missouri believe led to their own job losses.

The next big contest comes on March 15, where Florida, Illinois, Missouri, North Carolina and Ohio are up for grabs. Florida is one of the biggest prizes of them all, awarding 246 delegates to the winner. According to Real Clear Politics, Mrs. Clinton has an average lead of 25.6 percentage points in the early polls taken there. Mr. Sanders is expected to do well in the Midwestern states.

“Next Tuesday is the most important night for our campaign to date,” Mr. Sanders said in an email to supporters after his surprise victory in Michigan. “Five large states vote, and we have all the momentum. And what we’ve shown is that when we come together, we have what it takes to overcome what was once thought” inevitable.

Before Michigan, Mr. Sanders’s wins in eight of the first 20 Democratic contests had come from smaller and overwhelmingly white states. Mr. Sanders also had done well in caucus states like Iowa, which reward grass-roots support and enthusiasm. But in order to expand his coalition to compete in states like Arizona and California, Mr. Sanders will need to attract a higher percentage of the minority vote.

Despite her loss in Michigan, Mrs. Clinton has a large delegate lead Mr. Sanders is unlikely to overcome. Mrs. Clinton has dominated the black vote, winning eight out of 10 voters in a variety of states. She swept the South, where the black turnout was high — and they often can be a majority of the Democratic electorate — by largely promising to protect and extend President Obama’s legacy.

Mrs. Clinton also has a commanding lead with so-called “superdelegates,” officeholders and party officials free to vote for whomever they wish.

The Clinton camp brushed aside her stunning loss to Mr. Sanders in Michigan, and pushed the narrative the Democratic presidential contests will effectively end over the next several weeks. The team reiterated their strategy has always focused on the delegate race, and that her loss in Michigan was so narrow as to make little difference in the contest for delegates.

“From the beginning we have approached this nominating contest as a battle for delegates, so while Sen. Sanders has placed big bets on pulling out wins in individual states, we have sought to compete everywhere, in every state, with the goal of amassing the most delegates possible,” said Robby Mook, Mrs. Clinton’s campaign manager, in a conference call with reporters Wednesday.

“We are nearing the point where the delegate lead effectively will become insurmountable,” Mr. Mook said. “Even in a scenario where Sen. Sanders were to win all three of the states he is targeting in the Midwest … we would still expect to win significantly more delegates on the 15th based on [not] only our strong showings in those same three states, but our performance in the very delegate-rich states of Florida and North Carolina.”

Even though Mr. Sanders won Michigan, since the state awarded the delegates proportionally, Mrs. Clinton took home 58 delegates to Mr. Sanders‘ 65. And overall, Mrs. Clinton won the night because she dominated Mississippi, taking home 18 more delegates in aggregate than Mr. Sanders.

• Ben Wolfgang and S.A. Miller contributed to this report from Washington.

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