The Mamdani administration responded to concerns about the bumpy rollout of its second-home tax on Tuesday, stating that it had reduced the rolls of property owners who may owe the tax by over 4,000 and had extended the deadline for homeowners to prove New York City residency.

The city’s statements, offering its most detailed defense yet of the complicated process of administering the tax, came in a court filing in a lawsuit brought by property owners that argues that the city mishandled the introduction of the tax.

Last month, the Finance Department notified about 17,000 property owners that they might be subject to the tax — a levy on high-end “pieds-à-terre” — unless they could prove that the dwelling was their primary residence.

The tax, passed in the state budget, would apply to condos and co-ops that are used as second homes with a market value of at least $1 million and one-, two- and three-family second homes with a market value of more than $5 million. It had been sought by Mayor Zohran Mamdani to fulfill a campaign pledge to raise taxes on the wealthy.

The lawsuit, filed on Staten Island, does not challenge the legality of the tax. Rather, it argues that the city sowed chaos by sending out the 17,000 notices without first determining that the property owners were not full-time residents, and by releasing a list of nearly one million “properties that may be subject to the charge” that included the owners’ names.

A judge, Wayne M. Ozzi of State Supreme Court, paused the rollout, but the city appealed, which had the effect of staying the order until an appellate court weighs in.

On Tuesday, the city offered some explanations and acknowledged the complaints over its messaging.

It said that 2025 tax data it recently received from the state established residency for about 1,200 property owners among the 17,000 who received notices and that the city Finance Department had approved the proofs of residency submitted by another 2,900.

It said it was still reviewing proofs submitted by 2,600 other owners. The city said it was sending new notices, based on the 2025 tax data, to 10,800 property owners informing them that they need to prove residence. And it said that it was extending the deadline to file proof of residence to Oct. 6; it had earlier been Sept. 18.

The city largely defended having sent notices to property owners that said they could be subject to the tax unless they proved residency. “What petitioners characterize as improper burden shifting is actually an intentional design of the law,” the city wrote. “Petitioners may dislike or feel burdened by” the back-and-forth process — in which the city identifies properties as possible second homes, the owners submit proof of residency and the city accepts or rejects it — “but that does not make it illegal.”

Randy Mastro, a lawyer for the property owners who brought the suit, called the city’s filing “an admission that this was a massive screw-up.” He said the city had to “do a do-over” and “tell thousands of New York City homeowners who should never have received threatening notices in the first place that the city now recognizes that they live in their own homes.”

He said that the city should have used the 2025 tax data as the basis of the first round of notices, rather than 2024 data. (The city said it had sent those notices based on tax information available at the time.) And he said that given how many people will not file their 2025 taxes until the final deadline of Oct. 15, the city should have extended the deadline to prove residency to Oct. 16 rather than Oct. 6.

“The lawsuit continues because there are thousands more who are still in limbo,” Mr. Mastro said.

The city is seeking dismissal of the suit on several grounds. It says that the plaintiffs had not shown that the rollout had meaningfully harmed them and that proving that a property was a primary residence was straightforward, even when a property is owned by a trust or corporate entity.

It was not clear how many property owners in total would be taxed, or whether the exemptions would affect the total annual revenue that the tax is projected to raise, about $500 million. The state did not respond to a request for comment on projected revenue.