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In a winding speech short on specifics, and with none of his temporary post-Iowa contriteness, Donald J. Trump addressed a few thousand people at a Clemson University equestrian center here Wednesday night. It was his first appearance in South Carolina after a resounding victory in the New Hampshire primary.

“I had a choice between Mar-a-Lago in Palm Beach, or South Carolina,” he told the crowd on a frigid evening.

Before a largely white audience that included many veterans, older people and Clemson University students, Mr. Trump addressed his usual campaign issues: building a wall on the Mexican border, stopping crime committed by illegal immigrants, ending the Affordable Care Act, rebuilding the military, negotiating with drug companies on pharmaceutical prices, and promising, to a standing ovation, to ensure that America keeps “Christ” in Christmas.

He also criticized Hillary Clinton over the attack in Benghazi, Libya; Senator Bernie Sanders, whom he called a “wacky socialist”; and Jeb Bush, whom he termed “a schlepper” for his reliance on campaign spending of over $30 million only to finish fourth in New Hampshire.

“You have a guy like Bush,” Mr. Trump said of his frequent adversary, “who has this big fund. I’ll be honest, the last thing we need is another Bush. This guy. He says anything that’s on his mind. But you know, he’s got so much money. In New Hampshire, he spent $39 million and what is he, fourth or fifth, right? Think of it, $39 million.”

Though many of his comments reflected his standard stump speech, a seeming stumble over his stand on the Common Core education standards prompted Mr. Trump, who rarely acknowledges mistakes and who told the crowd he was operating on about an hour of sleep, to take to Twitter to try to explain himself.

During his speech, Mr. Trump strongly criticized Common Core, a federal initiative deplored by conservatives and some teachers’ groups, and assailed Mr. Bush for backing the program. But, at another point, when listing his proposals, Mr. Trump said: “We are going to do some things special. O.K., so you ready? Common Core, we’re gonna, we’re gonna keep; trade, we’re gonna make great; military, we’re gonna go through the roof, we’re gonna make it so great.”

Mr. Trump was asked to clarify his position on Twitter, where a recording of the comments had been posted. Mr. Trump wrote that “I have been consistent in my opposition to Common Core. Get rid of Common Core— keep education local!” He had been referring, he said, to Mr. Bush. “I was referring to the fact that Jeb Bush wants to keep common core.”

The moment did not seem to register at the event itself, however, and the largely jovial crowd often responded with applause.

Some in attendance, like Wayne Woolbright, 61, a nuclear safety consultant; his wife, Wanda, 53; and their 17-year-old son, Brad, whose birthday comes before the primary, displayed newly signed copies of Mr. Trump’s book “The Art of the Deal.” Mr. Woolbright said he believed that Mr. Trump’s skills as a negotiator would lead him to work across the aisle and reach compromises in Washington, and that he had a solid chance to win South Carolina.

Mr. Woolbright, a Pentecostal Christian, made a point of addressing what he described as unfair accusations that Mr. Trump held blanket hostility toward Muslims. “He is very open to all religions, as long as it’s not got to do with killing the other religions,” he said. “I honestly believe in my heart that he’s going to protect Muslims who are not practicing jihad, and I believe he’d protect any other religions. I believe he’d come down on Pentecostals if we were going to go kill Catholics.”

He added that the criticism leveled against Mr. Trump from people who have questioned the candidate’s religious bona fides — his ownership of casinos and his multiple marriages — was unfounded.

“Let he who is without sin cast the first stone,” Mr. Woolbright said. “If there’s a candidate out there that has no sin, I will change my vote from Mr. Trump to the other candidate. Because that person would be Jesus Christ. He’d be my first candidate. But he’s not running, so it’ll have to be Trump.”

Not everyone in the crowd was as enamored of Mr. Trump. Vialana Vazquez, 17, who also will be old enough to vote by the primary, came to the event with her father, Nelson Vazquez, 59, an occupational therapist. Mr. Vazquez, who has lived in Puerto Rico, New York and Florida, now lives in Clemson, S.C.

The prospect of a Trump presidency worries Ms. Vazquez, who considers Mr. Trump a racist. “He’s a really big hypocrite, and he’s rude,” she said, adding that she was leaning Democratic. “He’s just not a nice person, and his ideas are just not like mine at all.”

Her father also was not a Trump fan. “I’m a Rubio supporter. I’m not going to support Trump in any way,” Mr. Vazquez said. “But since we live a few miles away, I thought it would be good to see him. Years from now, we’ll look back and say, ‘Remember when Trump ran and he thought he was king of everything?’ ”

Roy Welch, 19, an African-American student at Clemson University, attended the event with two friends, Paul Vincent, 20, and Elizabeth Hill, 19, both of whom are white.

“I just don’t believe a lot of things he says,” Mr. Welch said of Mr. Trump. “And I personally think he’s obsessed with himself.”

“Well, you have to be a little bit vain to run for president, though,” Ms. Hill said.

Mr. Welch said that he and Ms. Hill had discussed racist accusations leveled at Mr. Trump. None of the three said they believed that Mr. Trump was a racist. “He’s very passionate about what he wants for the country, but it just comes off a little strong,” Mr. Welch said.

While Mr. Welch said he will not be voting for Mr. Trump, Ms. Hill said she would consider it. “I grew up watching ‘Celebrity Apprentice,’ ” she said. “We loved it. Seeing him, it’s the same exact personality as on the show. Him being, like, the boss.”