But when they took their places alongside the governor for the signing, most of the city elected officials had not been given a copy of the emergency order. Now, they are scrambling to decipher how the executive order might affect the city’s finances and how the city will fund repairs to the largest public housing system in the country.

The order by Mr. Cuomo was an unprecedented intervention in the agency’s 84-year history, one the governor had been hinting at for days before, as he escalated his criticism of the agency’s performance under Mr. de Blasio — the latest clash in the fractured relationship between governor and mayor.

At a news conference two days later, Mr. de Blasio voiced his frustration. “I don’t see how you write an executive order like this without having a consultative process with the city,” he said. Last week Mr. de Blasio said he “did not expect the hits from Albany to be as bad as they ended up being” and that Nycha would “have a serious price tag attached” in this year’s budget.

Before the governor’s mandate, the city had no legal obligation to fund Nycha, according to experts and city officials. The housing authority, like other public authorities in New York, is a state-chartered entity, though the mayor appoints its seven-member board. Its budget is not part of the city’s budget, and it relies primarily on dwindling federal subsidies and on rental payments from the system’s 176,000 units.

In recent years, the city has nonetheless chosen to voluntarily fund Nycha in response to the authority’s growing deficit, investing $3.7 billion in the agency since Mr. de Blasio took office, city officials said.