When Kelly Ayotte jumped into the New Hampshire governor’s race last year, she seemed to have clear advantages.

Ms. Ayotte, a Republican, served as a United States senator from 2011 to 2017 and had much better name recognition in the state than Joyce Craig, the former mayor of Manchester who became her Democratic opponent. Ms. Ayotte had also been the state’s attorney general, and was an adept fund-raiser. The popularity of the outgoing Republican governor, Chris Sununu, would also help a lot, the thinking went.

So far, though, none of that has been enough to give Ms. Ayotte a decisive edge in the race. Polls have shown the candidates locked in a tight race, with no clear leader. In interviews, some analysts and voters said that the explanation may lie in Ms. Ayotte’s record on abortion, and in aggressive advertising by Democrats suggesting that Ms. Ayotte might seek to restrict access in New Hampshire. Strong support in the state for Vice President Kamala Harris could be another factor.

Ms. Ayotte voted as a senator to defund Planned Parenthood and helped guide Neil Gorsuch, the conservative Supreme Court justice, through his confirmation process in 2017. She has also expressed support for the court’s decision in 2022 to eliminate the constitutional right to abortion — in which Justice Gorsuch sided with the majority — and turn the issue over to the states.

New Hampshire bans abortion after 24 weeks of pregnancy, with a few exceptions. Ms. Ayotte has said she would not entertain changes to the law, but some voters remain worried.

“She has said the law won’t change, but there is no guarantee,” said Marie Mulroy, 77, an independent voter in Manchester who has not yet decided between Ms. Ayotte and Ms. Craig. “I think there is distrust among women.”

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