But representatives of both the Democrat-led Assembly and the Republican-led Senate said they would continue to try to find agreement on a complete budget.
“We are going to continue to work,” said Carl E. Heastie of the Bronx, the speaker of the Assembly, addressing the news media amid dozens of fellow Assembly Democrats.
The state has been operating under an emergency budget since Monday, when the Legislature passed a so-called extender budget to keep the government funded, as well as authorizing billions in spending on infrastructure projects favored by Mr. Cuomo.
On Tuesday evening, the Senate began passing budget bills — six in total — and on Wednesday, the Assembly followed suit. Late Tuesday, the Assembly said there was also apparent agreement on the issue of raising the age of criminal responsibility to 18 from 16, a policy issue that had proved to be a point of friction.
But in a sign of the ups and downs of the budget season, during his briefing — the governor’s first appearance in Albany since the April 1 deadline passed — Mr. Cuomo said that ideological questions remained, surrounding whether corrections officers or the staff of the Office of Children and Family Services would monitor youthful offenders who had served sentences under the plan.
Another holdup in the negotiations centered on charter school funding, and the imminent expiration of a law that froze tuition increases for charters. If lawmakers do not act, charter schools will automatically reap a tuition increase, paid by the school districts, of $1,500 per pupil when the law expires in June. The increase would cost New York City about $200 million next year, Mr. Cuomo said.
Republicans who control the Senate support increasing tuition for charter schools, while the Democrats who rule the Assembly are fighting those increases, not wanting to divert money from traditional public schools.
“I think $1,500 is a windfall for the charter schools,” Mr. Cuomo said. “I understand how it happened. But it’s a torrent of funding.”
The final issue involved 421-a, the lapsed tax abatement program for developers meant to spur affordable housing. A dispute focuses on whether rent control, which lapses in two years, should be linked to the program’s renewal.
Mr. Cuomo also said he was seeking flexibility in the budget to allow him to decrease spending on Medicaid in the event that federal funding to the state is cut when the Trump administration unveils its plan in late May.
“It’s very important to me that we not put our financial feet into cement,” the governor said during an hourlong briefing in the ceremonial Red Room in the State Capitol.
Mr. Cuomo suggested that the budget could remain a moving target throughout the year, citing the federal idea of a “continuing resolution,” which allows for budgetary changes. “I am looking for continuing financial flexibility in the budget process,” he said. “But certainty and permanence in the operations that need to continue which is what we accomplished in the extender.”
The budget process has dragged on past its April 1 deadline, resulting in the tardiest plan of Mr. Cuomo’s tenure, including a fruitless weekend session that the governor called “a grace period.”
The continuing delays seemed to further agitate an exhausted Legislature.
“People miss their families, they have no clothes left, and nobody knows what’s going on,” said Assemblywoman Nicole Malliotakis, a Staten Island Republican.
Mr. Cuomo said he wanted the Legislature to continue working — “I hope they will stay until it’s done,” he said — while he seemingly floated the possibility that the negotiations could continue for the foreseeable future.
“I will stay here as long as they stay here,” he said.
The holdup in the budget has had a practical impact on lawmakers: Under state law, they receive their salaries only if a state budget is in place. (Mr. Cuomo is still getting paid — he earns $179,000 a year, and Wednesday was payday in Albany.)
Money may also have played a part in the current behind-the-scenes tension between the Legislature and Mr. Cuomo; last fall, the governor was effectively responsible for denying lawmakers a raise when his appointees to an independent pay commission refused to authorize an increase from their current base salary of $79,500.
On Wednesday, Mr. Cuomo said he did not believe “a relationship issue involving me” had contributed to the late budget.
Whatever the cause, the delayed budget has been a blemish on the governor’s streak of passing six budgets on time or close to it, though he still defended his performance.
“Am I disappointed? I’ll take my record — one out of seven,” he said, adding, “I don’t think any governor in modern history has gotten more on-time budgets.”
But exactly how late the budget would end up being was still an open question on Wednesday.
Harry B. Bronson, a Democratic assemblyman from Rochester, said that school aid was especially critical to many lawmakers because it was a hot topic in their districts, and that the exact education funding was still unclear.
And while he, like everyone else in Albany, said a deal was close, it was not yet closed.
“We’re in a place where if decisions get made, we can put it together very quickly,” Mr. Bronson said. “But that’s an if.”