Manhattan prosecutors are urging the judge who oversaw Donald J. Trump’s criminal hush-money trial to uphold his conviction, seeking to cast doubt on the former president’s long-shot bid to overturn the case because of a recent Supreme Court ruling.

In a court filing made public on Thursday, the Manhattan district attorney’s office argued that the Supreme Court’s decision this month granting Mr. Trump broad immunity for official actions he took in the White House had “no bearing on this prosecution.”

Although the high court’s ruling was a blow to a separate criminal case against Mr. Trump in Washington, the Manhattan charges did not hinge on official acts. Instead, the Manhattan prosecutors noted, he was convicted in May of covering up a sex scandal that had threatened to derail his 2016 campaign, a personal and political crisis that did not involve his conduct as president.

Mr. Trump’s lawyers, seeking to link the two cases, have mounted a novel argument. In a recent filing to the judge who presided over the Manhattan trial, Juan M. Merchan, they contended that the Supreme Court’s decision had invalidated at least some of the evidence presented in Manhattan, including the testimony of former White House employees and tweets that Mr. Trump sent as president. The Supreme Court, they noted, had held that official acts could be inadmissible as evidence — even if a case concerned private misconduct.

But the district attorney, Alvin L. Bragg, fired back this week, arguing that the former president’s lawyers had missed their window of opportunity to raise the immunity defense and then distorted the Supreme Court’s ruling once it emerged.

Prosecutors from Mr. Bragg’s office argued in the court filing that the ruling did not apply to the type of evidence they had deployed against Mr. Trump, and highlighted the personal nature of testimony that had nothing to do with Mr. Trump’s duties as president.