Last summer, the rising influence of the Democratic Party’s progressive wing in New York seemed almost boundless.

Progressive activists helped knock off an incumbent congressman, fueled upsets in several state legislative races and pushed policies on taxation and policing that put an anxious business community further on edge.

Next year, the movement may face its sternest test in the New York City mayoral race, a wide-open contest that will be the city’s most momentous in decades.

New York officials and strategists across the ideological spectrum say that the Democratic electorate has plainly shifted to the left in recent years, and a unified liberal front helped make the difference in a number of high-profile congressional and legislative races in the city and around the country.

But at a time of extraordinary economic crisis, staggering public health challenges and rising gun violence, the mayor’s race may serve as a barometer of whether the electorate will be swayed more by bold, progressive ideas or evidence of managerial competence — or whether they believe a single candidate can deliver both.

The challenge for progressive leaders will be to try to replicate their successes — best exemplified by Representative Alexandria Ocasio-Cortez’s shocking win in 2018 — to a citywide race with more than 3.7 million registered Democratic voters, in a political landscape where more traditional political gatekeepers still hold influence.

“We have an opportunity to really radicalize and get people behind a lot of the things that we’ve been talking about for a very long time,” said Tiffany L. Cabán, a progressive candidate who nearly won the Queens district attorney race last year and is now running for City Council. “What’s at stake here is the opportunity in this moment to have a mayor that is going to say that this is not about safe, small, incremental change that tinkers around the edges.”

The progressive push fell short in the 2018 Democratic primary for governor, when Gov. Andrew M. Cuomo decisively defeated Cynthia Nixon, the candidate of choice for many left-leaning organizations and leaders. Nor was it quite sufficient to avoid Ms. Cabán’s narrow defeat, or to win some contested House contests.

Some Democratic leaders argue that the ideas that excite young progressives have not always resonated in older, working-class communities of color across the five boroughs. The mayoral primary in June will test whether any candidate can bridge that divide.

“The socialist left is on the rise, particularly in neighborhoods where Black and Latino residents are being gentrified out of existence,” said Representative Hakeem Jeffries, the chairman of the House Democratic Caucus, who represents parts of Brooklyn and Queens and may become the first Black House speaker. “To the extent the success of the socialist left is in part tied to gentrifying neighborhoods, it remains to be seen how that will impact a citywide race.”

How left-wing activists and organizations will choose to wield their influence is unclear. Were all the groups affiliated with the progressive movement to align behind one candidate, they could have a sizable impact on the race.

So far, they are not coalescing.

“There’s a big question of whether folks do,” said Jonathan Westin, the executive director of New York Communities for Change. “I think the candidate that is able to cobble together all of those groups is the candidate that is going to win.”

The New York City Democratic Socialists of America has endorsed six candidates for the City Council, a move that promises significant organizational assistance. But it has yet to make an endorsement in the mayoral race, and several people affiliated with the organization do not expect it to.

“If we had a mayoral candidate who came from the D.S.A., I think that would have been one thing,” said Susan Kang, a D.S.A. member and a professor at John Jay College of Criminal Justice. “We’re trying to be very strategic in how we use our labor.”

Another complicating factor is the popularity of Scott Stringer, the city comptroller and a leading mayoral candidate, among some prominent younger progressive lawmakers. In 2018, Mr. Stringer endorsed a D.S.A. stalwart, Julia Salazar, in her race for State Senate over the incumbent, Martin Dilan. Ms. Salazar won her race, and Mr. Stringer won her endorsement for mayor, along with several other high-profile endorsements from progressives.

Mr. Stringer has also won the backing of a few key unions, including most recently the Communications Workers of America, an early supporter of Mayor Bill de Blasio.

“Some people are a little bit disappointed that the current progressive front-runner is a white guy and certainly not an insurgent in terms of his background,” said Michael Kinnucan, a New York City D.S.A. member.

Nor is it clear whether several other progressive groups, including the Working Families Party, will play a role in the primary.

“We see ourselves as coalition builders, aligning the left, aligning working people’s institutions behind a candidate, a movement or a set of issues that can help shape a much stronger landscape for working people in New York City,” said Sochie Nnaemeka, the party’s state director.

Ms. Ocasio-Cortez offered her endorsement in a number of congressional and state primaries earlier this year, and a number of the mayoral candidates would probably covet her backing. A spokeswoman did not respond to a request for comment about Ms. Ocasio-Cortez’s endorsement plans in the race.

Even if New York progressives don’t unite behind a single candidate, they are already affecting the discourse of the race, as even candidates like Raymond J. McGuire, a longtime Wall Street executive, sound increasingly open to higher taxes on the wealthy.

But some traditional New York City power brokers are skeptical of a fiercely ideological pitch in this race, when city residents face so many tangible challenges.

“People are a little bit beleaguered when it comes to all of these ideological fights,” said Michael Mulgrew, the president of the United Federation of Teachers. “It’s more, ‘OK, who can start to steer this ship toward a better horizon?’”

The upcoming primary will also probe the citywide appeal of progressives’ language and policy proposals after their success in a series of more local races.

For example, there is evidence that in some poor and middle-class communities of color, slashing funding for police, a major left-wing priority, is controversial. That’s an issue that has divided the mayoral field.

Eric Adams, the Brooklyn borough president and a former police officer, and Mr. McGuire have both made overtures to the donor class while stressing their appeal to working-class Black New Yorkers. Both are betting that a citywide race will draw a diverse range of voters who do not all share the most far-reaching goals — defunding the police or imposing more taxes on the wealthy, for example — of prominent progressive organizations.

“I’ve never walked into one meeting, one civic group, one block association, one NYCHA development meeting where someone said to me, ‘I want less cops on my block,’” said Mr. Adams, who ran a police reform organization while at the New York Police Department. “Just the opposite: ‘Where are my police? What are they doing?’”

Several of the candidates are seeking to present themselves as the right blend of visionary progressive and seasoned administrator — perhaps none more so than Mr. Stringer, who has promised to “manage the hell out of this city” as he also seeks to rack up a list of endorsement from left-wing leaders.

He dismissed concerns that progressives might not want to elect a white man at this moment in history, noting he is the only candidate to have won citywide office and pointing to the racially diverse coalition supporting him.

“I don’t think I would be attracting this very powerful coalition if I was in simply the lane of what I look like,” he said.

Councilman Carlos Menchaca, of Brooklyn, and Dianne Morales, the former nonprofit executive, are running among the most progressive campaigns in the race. Asked whether she had spoken with key left-wing organizations about a possible endorsement, Ms. Morales said “beginnings of conversations” were underway, though she declined to specify which groups she was talking to.

“I have been on the ground as an organizer and activist,” she said. “My candidacy in particular is one that speaks to kind of mobilizing and organizing on the ground.”

Mr. Stringer said he had yet to reach out to the D.S.A. about an endorsement. Mr. Menchaca said he would welcome the support of any organization that wants to help him “turn the page” on the de Blasio era.

Mr. Jeffries suggested that in a time of deep crisis, a candidate with a more pragmatic message may have an edge. He also made a point to speak highly of incoming Rep. Jamaal Bowman, who, boosted by leading progressive groups, defeated Representative Eliot L. Engel last summer in a district that covers parts of the Bronx and Westchester County. Mr. Jeffries had backed Mr. Engel.

“The person who rises to the occasion of a forward-looking, progressive attainable vision is the mayoral candidate who is likely to prevail,” he said.