It also presented a chance to double down on his credentials and position himself as a progressive leader for New York — and perhaps the nation — as he faces a primary challenge from the left and a crescendo of voters across the country, at least within his own party, who seem to have little appetite or patience for moderates.
The governor’s supporters argue that Mr. Cuomo merely feels that he must fix problems unattended to by the mayor and city officials. The mayor, not surprisingly, sees it differently, often saying Mr. Cuomo is meddling in city affairs.
City Hall offered a muted reply to the budget on Saturday, with Eric Phillips, the mayor’s press secretary, saying that the budget “appears to respond to the mayor’s demands on behalf of the city’s straphangers,” in reference to plans that would protect subway funds and provide long-term revenue.
“There are no excuses left for the governor to hide behind,” Mr. Phillips said in a statement. “He must do his job and fix the subways.”
“Governor Cuomo’s obsession with Mayor de Blasio is great when it means the governor lives up to his promises to public housing tenants,” he added. “The governor’s obsession is a problem when his focus is on adding bureaucracy and slowing down our efforts on schools and corrections reform.”
The budget passed before dawn on Saturday morning, just hours after Mr. Cuomo announced an agreement with Albany’s legislative leaders.
While budgets often are concerned with the minutiae of state operations, Mr. Cuomo’s plan singled out some name-brand, nationally known locales — the subways, for one — which the governor said the city must help pay to repair or suffer hundreds of millions in withheld funds.
Some city officials have rejected Mr. Cuomo’s call to pay for half the repairs, arguing that 70 percent of the operating budget of the controlling agency, the Metropolitan Transportation Authority, already comes from New York City payroll and other taxes. And Mr. de Blasio rarely misses an opportunity to point out that the state, not the city, controls the subway system.
The subways are also a key element in the subplot that many observers see undergirding the governor’s current fury with City Hall: the emergent candidacy of the actress Cynthia Nixon as a primary challenger to Mr. Cuomo.
Ms. Nixon, a close ally of Mr. de Blasio who announced her campaign as negotiations on the budget heated up in mid-March, has made no secret of the fact that she intends to attack the governor on the state of the subways. (Her campaign website has a section devoted to #CuomosMTA.)
She has already called Mr. Cuomo’s liberal credentials into question. At her campaign’s debut, she asked if he was a “real Democrat,” and she has linked him to the Independent Democratic Conference, a group of eight breakaway Democrats in the State Senate who collaborate with Republicans. Several progressive groups in New York that have expressed their enthusiasm for Ms. Nixon’s candidacy have suggested that they will take their anger at President Trump out on the I.D.C. and its allies.
The budget seemed a perfect opportunity for Mr. Cuomo, who has spent most of the last 40 years involved in New York State politics, to show his mastery of Albany’s levers of power — and his knowledge of how to use them against his foes.
“Cuomo never, ever missed an opportunity to stick it to de Blasio,” said Douglas Muzzio, a professor of political science at Baruch College.
In the spending plan, Mr. Cuomo also sought and won new oversight of the city’s public schools. The budget requires districts located in cities with populations of more than one million to submit a detailed, school-by-school accounting of how they spent state educational funds. Previously, the state allocated money to districts but did not oversee the distribution of that money.
Like the city’s subways, if districts do not comply, the state can withhold funding.
Mr. Cuomo billed the move as a much-needed increase in transparency. But it also served a dual political purpose: It jabbed at the tradition of mayoral control of New York City’s schools, and it also allowed the governor to take an aggressive stand on the issue of educational equity — a marquee issue of Ms. Nixon, a longtime education activist who has placed the issue squarely at the heart of the early stages of her campaign.
Specifically, Ms. Nixon has blasted Mr. Cuomo for underfunding schools, accusing him of neglecting children of color in the poorest districts. But Mr. Cuomo on Friday said the issue was not the amount of money being spent, but rather how it was being spent.
The deteriorating conditions at the New York City Housing Authority, the most recent flash point in Mr. Cuomo and Mr. de Blasio’s long-running rivalry, also took center stage in Friday’s budget presentation. In addition to promising $250 million in new funding for the authority, Mr. Cuomo said he would issue an executive order to appoint an independent monitor, to be selected by the City Council in collaboration with tenants.
Mr. Cuomo has visited public housing complexes in the city three times in the past few weeks, always with cameras and supporters in tow, blasting what he has called the authority’s mismanagement and abuse of tenants. His first visit, to a complex in the Bronx on March 12, took place when Mr. de Blasio was out of town.
For its part, the mayor’s office has said that the state has been slow to deliver $200 million in already-promised funds.
In his open feud with Mr. Cuomo, Mr. de Blasio has few allies in Albany, the state capital, where he also has alienated Republicans who rule the Senate with past campaigns to oust them. His single consistent bastion of influence is in the Assembly, where Democrats from the city dominate and where Carl E. Heastie, the chamber’s speaker, usually acts as a bulwark against policies that seem to overreach into city affairs.
Gerald Benjamin, a professor of political science at the State University of New York at New Paltz, said that the two men’s political ambitions, Mr. de Blasio’s solid bona fides with progressive voters and Mr. Cuomo’s reputation as a centrist had combined to make their relationship especially fraught.
“I don’t know if they like each other,” he said. “They appear not to.”
On Friday night, the reaction to the governor’s plans to use state control to devise plans for the Penn Station area, coveted by developers (including some who have been generous donors to Mr. Cuomo’s campaigns), was swift and angry.
“It is wrong for the governor to try to take over urban planning, traffic management and real estate development in New York City,” said Richard Gottfried, the veteran Democratic assemblyman who represents the area. “That’s what this bill is aimed at.”
In the Senate, too, downstate Democrats expressed alarm at the budget’s perceived power grab in the city. Senator Liz Krueger, a consistently liberal voice who represents a district on the East Side of Manhattan, said during debate on the Senate floor that the bill signaled the death of home rule.
And Assemblywoman Catherine Nolan, another veteran Democrat who chairs that chamber’s education committee, also predicted that the governor’s decision to allow his Division of the Budget to oversee and potentially limit the spending of big school districts, such as New York City’s, could backfire.
“Districts statewide are going to deeply resent the governor’s ability to withhold funds so arbitrarily in such a punitive fashion,” said Ms. Nolan, who — for the first time in 33 years in Albany — said she voted no on a major part of a budget bill. “Never underestimate the grass-roots power of local control.”