Could Democratic presidential candidate Dean Phillips get enough votes in Tuesday’s New Hampshire primary to — not necessarily win — but embarrass President Biden, whose name will not be on the ballot and who hasn’t campaigned in the state? Could this open the door to Mr. Phillips teaming up with No Labels for a third-party bid?

Biden campaign surrogates flocked to the Granite State this weekend to make sure that doesn’t happen. Congressman Ro Khanna and Boston’s mayor, Michelle Wu, joined Democratic leaders at “Write In Biden” events across the state, where volunteers held signs in the cold for those driving by.

A three-term congressman who announced his presidential bid just three months ago, Mr. Phillips has low name recognition and is not gaining much traction. He is polling on average at only 11 percent in New Hampshire, according to Real Clear Politics.

Yet an Emerson poll from earlier this month shows Mr. Phillips with 16 percent, and an American Research Group poll released Sunday shows Mr. Phillips at 32 percent to Mr. Biden’s 54 percent. This latter poll is an outlier — American Research Group polling on the Republican side also shows a much closer race than other polls — but Democrats are worried that low turnout and frustrations with the president for ignoring the state could backfire on Mr. Biden.

“I support Biden but I’m not voting for him on Tuesday,” a registered Democrat from Nashua, Bob Lamay, told the Sun at a Phillips rally over the weekend. “Why would I write in Biden’s name if he’s not even going to show up?”

Mr. Lamay said he used to work as a vote registrar, and he’s upset with Mr. Biden for creating so much extra work for registrars who will have to manually count the write in votes. “There are quite a few people upset about that,” he said.

“If he wrote you off, why would you write him in?” Mr. Phillips told a packed room at a senior center in Nashua Saturday.

Mr. Biden’s name will not be on the state’s primary ballot because New Hampshire refused to comply with the Democratic National Committee’s decision to put South Carolina’s primary first in order to, in the words of Mr. Biden, give Black voters “a louder and earlier voice in the process.” As a result, Mr. Biden has declined to campaign in the state, and the Democratic National Committee wrote a letter to the state’s Democratic Party this month saying the contest will be “meaningless” and that the results will not be used to award delegates.

Even if the primary is “meaningless,” the write in campaign is working to avert an embarrassing outcome. If Mr. Biden gets less than 50 percent of the vote as an incumbent, it would highlight his low approval rating and could renew calls for him to drop out of the race. Lyndon Johnson, an incumbent president, did just that in 1968 after winning in New Hampshire with just 50 percent in a write-in campaign.

“Mr. Biden took the Granite State for granted,” Mr. Phillips said. “I think the country would be better served with a Haley-Phillips matchup.”

Mr. Phillips’ pitch to voters is centered on the fact that a majority of Americans don’t want another Biden-Trump race. He repeats that fact often in his speeches, appealing to what he called on Saturday the “exhausted majority of America.”

Aside from some far-left proposals like supporting Medicare-for-All, Mr. Phillips is trying to appeal to the “commonsense” middle. A former member of the congressional Problem Solvers Caucus, he said Saturday that he would not rule out a No Labels run. “It would have to be a Joe Biden-Donald Trump rematch that shows Joe Biden is almost certain to lose,” he told the Times.

Mr. Phillips is also attacking the Democratic Party for treating the presidential primary like a coronation and disenfranchising New Hampshire’s Democratic voters. To many Granite Staters, who prize their outsized role in presidential politics, this message is resonating.

“If they say New Hampshire’s primary election doesn’t count, I won’t vote for a Democrat this general election. I might never again,” a mechanical engineer and registered Democrat from Dover, Jason Provencher, tells the Sun. “I will probably vote independent.”

Mr. Provencher, though, says he’s not voting in Tuesday’s primary. “Biden is going to win the Democratic primary anyway so there’s really no point,” Mr. Provencher said. He didn’t know who Mr. Phillips was. There will be 21 names on the Democratic primary ballot. With little name recognition, that could work against Mr. Phillips.

The Sun spoke with two dozen registered Democratic voters in the state. A handful said they weren’t voting in the primary because it doesn’t count and Mr. Biden is going to win anyway. Some were confused about the write in campaign. Most knew nothing about Mr. Phillips.

“I just don’t think he has the name recognition or bona fides to be president,” a retired funeral director and registered Democrat at Portsmouth, Ken Walsh, tells the Sun. He is going to write in Mr. Biden but says he “might be interested” in a moderate third-party ticket by No Labels.

While New Hampshire’s secretary of state, David Scanlon, predicts record turnout in the Republican primary Tuesday, he said this week he anticipates a low turnout of only 88,000 on the Democratic side. Nearly 4,000 Democrats switched their party registration to “undeclared” or Republican before the October deadline. Undeclared voters can cast a ballot in either party’s primary. Governor Haley is counting on a large showing from independent, anti-Trump voters if she has any shot at winning.

“We definitely have neighbors who are Democrats who, before whatever the deadline was, they went and registered as independents, and they’re going to vote for Nikki,” a realtor and registered Democrat at Dover, Sara Treacy, tells the Sun.

Ms. Treacy says she is going to write in Biden. “We have gotten a thousand postcards saying ‘write it in,’” Ms. Treacy says. She says she’s “not interested” in learning about Mr. Phillips.

At the Phillips event Saturday, a supporter donning a “Dean Phillips for President” tee shirt, Anne Morris told the Sun she “didn’t even know [Mr. Phillips] was running” until three weeks ago. “That’s what most people think. They think that Biden is the only person you can vote for on the Democratic side,” she said. “I’m talking to anyone who will listen to me about Dean.”

Like with Ms. Haley finally starting to attack Mr. Trump, though, it may be too little, too late.