Vice President Kamala Harris’ somewhat unexpected stop Wednesday in New Hampshire raised eyebrows and questions about whether her campaign is worried that Republican nominee Donald Trump could break the GOP’s more than two-decades-long losing streak in the state.

Despite history and polls showing Ms. Harris leading Mr. Trump in New Hampshire, the Democratic presidential nominee returned to the Granite State to unveil a small business tax proposal and distance herself from President Biden on capital gains taxes.

“If you earn a million [dollars] or more, the tax rate on your long-term capital gains will be 28% under my plan because we know when the government encourages investment, it leads to broad-based economic growth and it creates job, which makes our economy stronger,” Ms. Harris said.

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Mr. Biden had advocated for those taxpayers to face a 39.6% long-term capital gains rate. The current long-term rate is a maximum of 20%.

Ms Harris also promised to expand the start-up tax deduction for small businesses from $5,000 to $50,000. She pledged to provide low-interest loans to existing small businesses, reduce the regulatory burden and simplify tax filing.

“As president, one of my highest priorities will be to strengthen America’s small businesses,” Ms. Harris said.

Once again, the Harris event was tightly scripted. Her team outlined her small business plan with preferred media outlets ahead of time, and she delivered a speech on the topic while being surrounded by party faithful at a woman-owned brewery in North Hampton.

The Harris campaign has avoided unscripted moments that could hurt her image and threaten to jeopardize the momentum she has been carrying since she replaced President Biden atop the Democratic ticket in July.

The strategy will be tested on Tuesday in Philadelphia when Ms. Harris debates Mr. Trump for the first time.

The approach has worked so far, including in New Hampshire, where Ms. Harris holds a 5-point lead over Mr. Trump, according to polling averages that have hardened the perception that New Hampshire is hers to lose. George W. Bush, in 2020, was the last Republican presidential candidate to win the state.

“It is not a state secret that Donald Trump is down here in New Hampshire,” said Neil Levesque, director of the New Hampshire Institute of Politics at St. Anselm College. “We have done two polls here and there have been a slew of others that have shown this, and you haven’t seen the candidate or any top-tier [Trump] surrogates in the state for some time.”

Mr. Levesque said Ms. Harris’ visit allowed her to check a box.

“She needs to come to New Hampshire between now and the election, at least once, to say I was there and I care about your votes and New Hampshire is a state where people notice if you haven’t been here,” he said. “It also creates more enthusiasm for her workers.”

The Boston Globe reported this week that a Trump volunteer from Massachusetts sent an email claiming the campaign had thrown in the towel in the state, where four years ago, Mr. Trump lost by more than 7 points to Mr. Biden.

The Trump campaign dismissed the story. Mr. Trump and his team insist he still can win the state’s four electoral votes in November.

“I don’t even understand why it is close,” Mr. Trump said Wednesday on “Good Morning NH with Jack Heath,” a radio show and podcast in New Hampshire. “She’s a Marxist.”

Mr. Trump said he could not believe New Hampshire would support Ms. Harris after Mr. Biden ignored the state in the primary election cycle, because state officials refused to follow his plan to shake up the traditional nomination calendar by putting South Carolina first.

“They threw New Hampshire away and I don’t think they should get away with it,” he said.

In 2020, Ms. Harris did not focus much on New Hampshire during her initially mega-hyped bid for the Democratic presidential nomination. Instead, she bet that she would have better luck with the predominantly Black primary electorate in South Carolina.

The strategy blew up in her face; her campaign flamed out spectacularly. She was out of the race before the nomination contests began, but her name still appeared on the 2020 New Hampshire primary ballot.

She received 129 votes.

“As soon as people got to know her, she started rapidly disappearing,” Mr. Trump said Wednesday.