New York governor Andrew Cuomo and New York City mayor Bill de Blasio once again had differing opinions on who has authority to implement certain Covid-19 policies.

This time, it involved De Blasio’s announcement on Saturday morning that New York City’s school district, which oversees 1.1 million students, would shutter for the rest of the academic year to try to slow the spread of the coronavirus. De Blasio said his administration was forging a “comprehensive plan” to safely reopen schools in September and that “next year is going to have to be the greatest academic year New York public schools have ever had”.

De Blasio, like Cuomo a Democrat, said it was not an easy decision to close schools in favor of online learning, but that “it is the right decision, and it’s also a decision made a little clearer by the fact that the distance learning is working more and more every day”.

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However, at Cuomo’s daily press briefing shortly afterwards, the governor insisted “there has been no decision on [closing New York City] schools” and that the decision whether or not to close schools is “is my legal authority in this situation”.

Cuomo added: “When we made the decision to close the schools, we made it for the entire region. Any decision to reopen them will also be a coordinated decision.”

Cuomo was then pressed on the topic and De Blasio’s earlier remarks. “That’s his opinion. He didn’t close them and he can’t open them,” Cuomo said. “It happened on a metropolitan-wide basis and we’ll act on a metropolitan basis, coordinating with Nassau, Suffolk and Westchester [counties].”

Adding to the confusion, an email from the city to parents was issued while Cuomo spoke, advising of the extended school closing. “NYC school students will continue with remote learning for the rest of the 2019-20 school year,” it said.

Freddi Goldstein, De Blasio’s press secretary, took issue with Cuomo’s comments.

She tweeted: “The governor’s reaction to us keeping schools closed is reminiscent of how he reacted when the Mayor called for a shelter in place. We were right then and we’re right now. Schools will remain closed, just like how we eventually – days later – moved to a shelter in place model.”

Goldstein’s tweet alluded to that fact that Saturday’s dispute was the latest bout in a long-running grudge match between the two elected officials.

When De Blasio said last month New York City residents should prepare for a “shelter-in-place” order, Cuomo said the city didn’t have the power to make such a declaration. Days later, Cuomo announced a “New York state on pause” order, directing non-essential businesses to close and telling people to stay 6ft away from others in public. The order sounded much like shelter-in-place, a term De Blasio has continued to use.

School buildings in New York City, the center of the pandemic in the US, have been closed since 16 March. All school buildings in the state have been closed since 18 March following an executive order from Cuomo. The school closures were initially announced for a few weeks, back when the virus’s full impact was not known. A massive effort to move school instruction online has met with mixed success in the city, where many low-income students lack wifi and devices for connecting to their virtual classrooms.

Cuomo, whose national profile has risen during the pandemic, said again on Saturday he is not interested in running for president.

When a reporter said some Democrats would prefer Cuomo to former vice president Joe Biden as their party’s presidential nominee, Cuomo said: “That is on one hand flattering. On the other hand, it is irrelevant.”

“I have no political agenda. Period,” he said. “I’m not running for president. I’m not running for vice president. I’m not running anywhere. I’m not going to Washington. I’m staying right here.”

• The caption to the picture was amended on 14 April 2020 because an earlier version said Andrew Cuomo was on the right. This has been corrected.