The governor also appoints the chairman of the board, with four members recommended by the mayor and one each from the seven counties served by the commuter lines. (Four of the counties combine for one vote.) The members are confirmed by the State Senate.

Mr. Cuomo has taken an increasingly hands-on role at the authority and exerts influence on a wide range of issues, from pushing it to announce an improvement plan for the subways and deciding what to prioritize in the capital improvement plan. His office has issued a steady stream of feel-good announcements about the subways over the years, particularly around the time of the opening of the Second Avenue subway line.

“You know who runs the M.T.A.? The governor has the majority of members,” Mr. Cuomo said in December while touring the new stations. “The mayor has members, the county executive has members, but the governor has the majority of members. And if a governor wanted to step up and be responsible, he or she could be. And what I said is I’m going to step up and take responsibility.”

Those comments appeared to be at odds with the stance he took when speaking with reporters on Thursday, when he played down his influence over the authority and his role in its latest six-point plan to quell the flood of delays. “I have representation on the board,” he said.

It appeared to be those comments that had set off City Hall. As recently as last week, aides to Mr. de Blasio saw little point in having him publicly criticize the authority or the governor about the problems in the subways, where delays have spiked over the past five years. Doing so, the thinking went, could be counterproductive in areas where the city and state are working together productively, such as select bus service.

Many New Yorkers remain confused about who controls the city’s subway system. “New Yorkers who live in the city, they will believe that the M.T.A. is the city’s responsibility,” said Councilman Ydanis Rodriguez, a Manhattan Democrat and chairman of the City Council’s transportation committee who said he takes the subway to work almost daily.

That sense appeared to be borne out in a Baruch College/NY1 poll released on Thursday in which about a third of respondents blamed the mayor for subway problems, while roughly the same percentage blamed the governor and a third were unsure of who to blame.