In February, fresh off helping elect Zohran Mamdani as mayor of New York City, a retinue of democratic socialist officials met with Carl Heastie, the powerful speaker of the State Assembly, who had frequently tangled with Mr. Mamdani and his allies.

Mr. Heastie reiterated his long-held frustration with the group’s penchant for challenging Democrats in primaries and said he wished that they would stop targeting members of the same party, because it was not the best path forward.

The point, which came shortly after a rally D.S.A. members had convened in Albany to push for more taxes on the rich, was not taken to heart.

“We told him then, and we’ll tell him now: That’s not something we can do,” Grace Mausser, a co-chairwoman of the Democratic Socialists of America’s New York City chapter, said about Mr. Heastie’s request in the meeting, which has not previously been reported. “We have to continue to grow our numbers and grow our influence, and that means challenging incumbents.”

On Tuesday, Mr. Heastie and other establishment Democrats watched as D.S.A.- backed candidates won two House primaries in New York City, unseating an incumbent in the process, and also claimed several victories in state legislative primaries, including three over incumbents who lost to D.S.A.-endorsed candidates.

All told, there may end up being as many as 16 D.S.A.-backed legislators in Albany come January, and that doesn’t include elected officials who are aligned with their policies but were not on the organization’s slate of officially backed candidates.

“We have pretty much doubled our cohort in Albany, and leaders there are going to have respond to that,” said Gustavo Gordillo, another of the local D.S.A. chapter’s co-chairs, who was in the meeting with Mr. Heastie but declined to discuss its details. (A spokesman for Mr. Heastie confirmed the meeting had occurred but declined to comment.)

The state’s next legislative session does not begin until January, giving legislative leaders plenty of time to prepare for the possible influx of democratic socialists and whatever priorities they choose to push.

Kara Cumoletti, a spokeswoman for Gov. Kathy Hochul, pointed to the governor’s comments from January about not being worried about pressure from the left because, as the head of the state party, she has to consider a broad array of perspectives.

“There are over 140 Democrats serving in the State Legislature, and the governor has worked with all of them to accomplish many shared priorities,” Ms. Cumoletti said in a statement. “The only thing that drives her decisions is what’s best for New Yorkers.”

The most contentious issue is likely to involve higher or new taxes on the wealthy or high earners. Ms. Hochul has insisted that taxes — in contrast to an essential promise of Mr. Mamdani’s mayoral campaign — should not go up.

Yet the governor showed a willingness to compromise by unveiling a new levy on high-priced second homes in New York City, even though the tax, once it goes into effect, is unlikely to bring in as much revenue as what the mayor and his allies desire.

Though Ms. Mausser celebrated the so-called pied-à-terre tax, she made it clear that it was just a starting point on taxing the wealthy.

“I really hope that Governor Hochul is watching closely and understands that her strategy of obstinance and her orientation of ‘I don’t respond to pressure’ is not working and people can, and are willing to, vote out incumbents,” Ms. Mausser said.

With the D.S.A. recording victories across the state — a member won a primary in Buffalo — Ms. Hochul may need to determine how much to entertain the group’s demands amid an increasing pressure campaign. Having Mr. Mamdani play a more vocal role next year in the likely legislative fights over raising taxes could prove pivotal.

Speaking to reporters on Wednesday, the mayor sidestepped questions on whether he would use his growing influence to pressure the governor or legislative leaders to advance his agenda. At the same time, he reiterated his appreciation for Ms. Hochul’s willingness to prioritize resources for expanding child care in the city and his excitement about the potential new crop of legislators in Albany next year.

“What their successes represent is a shift in the balance of power between working people and special interests,” Mr. Mamdani said on the anniversary of his victory in the Democratic mayoral primary. “And finally, working people are going to have more of a voice in the halls of power, wherever that hall may be — be it Albany or be it in D.C.”

Some of the mayor’s most ardent supporters have questioned whether he should be more confrontational with Ms. Hochul. Earlier this year, for example, he skipped a Tax the Rich rally in Albany that Ms. Mausser, Mr. Gordillo and other D.S.A. members had organized and attended — even though the rally’s theme coincided with the mayor’s agenda.

Some also voiced dissatisfaction that Mr. Mamdani had refused to endorse the D.S.A.’s entire slate of legislative candidates, in apparent deference to Mr. Heastie.

Eon Huntley, a democratic socialist in Bedford-Stuyvesant, handily defeated Assemblywoman Stefani Zinerman on Tuesday. He said that his not having been endorsed by the mayor had not shifted his goals.

“I may not have been endorsed by him, but this campaign also was explicitly about delivering on the affordability agenda,” Mr. Huntley said.

Other candidates, like Eli Northrup and Brian Romero, were endorsed by Mr. Mamdani but not by the D.S.A.

A longtime legislative aide, Mr. Romero watched with disappointment in recent years as Ms. Hochul and legislative leaders rolled back environmental regulations and did not raise taxes to the level he deemed necessary.

After winning a primary for an Assembly seat in Queens on Tuesday, Mr. Romero said he was excited to work on the issues that Mr. Mamdani ran on last year as well.

“I absolutely see myself as part of the ‘Z Caucus,’” Mr. Romero said, using a nickname many of the mayor’s friends use for him. “We are ready and excited to advance his agenda.”

Patrick Jenkins, a lobbyist who once served as an aide to Mr. Heastie, said new members were going to have to figure out the complexities of Albany and how to advance their agenda.

“The rhetoric that you use during a campaign isn’t always reality,” Mr. Jenkins said. “You wind up having to negotiate with people that live in Buffalo and Rochester and Syracuse and you have to now balance limited resources with all kinds of other priorities.” Though the results are not final, a D.S.A.-backed candidate is running neck and neck for an Assembly seat in Syracuse.

The D.S.A.’s ability to shape policy in Washington still seems extremely limited, especially with Republicans in power. House Speaker Mike Johnson said on Wednesday that there were “mini-Mamdanis popping up all over the country.”

He added: “It is a dangerous thing. This is not a joke. We’re in a fight right now to save the Republic.”

The D.S.A.’s influence should be greater in New York, where the group’s leaders are mulling their legislative agenda and already thinking about future elections and greater ambitions.

In an interview, Ms. Mausser mused about a socialist running for president in 2028 and other tantalizing races that year, when Senator Chuck Schumer, the Democratic minority leader, is up for re-election.

“Perhaps a socialist running for Senate?” Ms. Mausser said.