The Homeland Security Department quietly canceled one of its major amnesty contracts after a judge’s ruling against the program last week, but officials have struggled to explain how they are complying otherwise even as the administration filed an appeal Monday asking that it be allowed to begin processing applications immediately.

Republicans on Capitol Hill searched for an honorable retreat from a fight over homeland security funding that could cast blame on them for a second partial government shutdown in 17 months.

Meanwhile, President Obama’s lieutenants struggle to figure out how to live up to Judge Andrew S. Hanen’s ruling.

In a notice posted Friday, U.S. Citizenship and Immigration Services said it had “hereby cancelled” a contract that would have hired hundreds of outside workers to receive and process amnesty applications.

USCIS refused to say whether it is still spending money to hire its own workers or prepare for other parts of the amnesty even after Judge Hanen ruled Feb. 16 that it ran afoul of the law.

The agency also has left immigrant rights groups struggling to understand what happens to so-called Dreamers, who are eligible for a 2012 amnesty but who were poised for more benefits under the expanded amnesty.

One immigration lawyer said the White House was trying to figure out how to comply with the judge’s ruling without demoralizing millions of illegal immigrants eagerly awaiting the chance to apply for the amnesty, fearing another disappointing legal defeat could cripple the program.

After a week of debate, the administration filed an appeal asking Judge Hanen for an emergency stay of his own injunction. The appeal said the administration wanted to start the expanded amnesty for Dreamers as soon as possible.

“This reflects our commitment to trying to work this through the legal process as quickly as possible,” said White House press secretary Josh Earnest.

The case likely will be sent to an appeals court in New Orleans and, eventually, to the Supreme Court.

The administration signaled that it is determined to begin approving at least some amnesty applications as soon as possible. The longer the process waits, the more agents in the field will be pursuing immigrants whom Mr. Obama will never deport anyway, the administration argued.

“If law enforcement organizations do not ensure that their limited resources are directed to their highest priorities, overall public safety might be compromised,” Customs and Border Protection Commissioner R. Gil Kerlikowske said in an affidavit.

The administration was so eager to approve some applications that it asked Judge Hanen to allow the amnesty to begin in at least a dozen states that have told the court they support the program. The request marked a striking turnaround for Mr. Obama, who just a few years ago took Arizona to court arguing that the state’s laws were creating a patchwork of the immigration system.

The 26 states that sued to stop the amnesty, led by Texas, told Judge Hanen that there should be no rush to lift his injunction. They noted that Mr. Obama’s attorneys waited a full week before filing their request.

“Indeed, if defendants had any compelling claim from a looming, irreversible harm from temporary injunctive relief, they would have featured it previously,” Texas Assistant Attorney General Angela V. Colmenero wrote in a memo to Judge Hanen.

She also pointed to Judge Hanen’s remark on the amnesty that “there will be no effective way of putting the toothpaste back in the tube.”

Judge Hanen’s ruling affected Mr. Obama’s amnesty for illegal immigrant parents with children who are either U.S. citizens or legal immigrants, and the president’s expansion of his 2012 amnesty for Dreamers, which lifted the age cap and allowed them to apply for three years of legal status and work permits to be able to compete legally for jobs.

Congress has refused to provide funding for the amnesties, which raises questions about how Mr. Obama was able to lay the groundwork while not running afoul of spending laws.

USCIS has declined to answer questions about its activities, though it acknowledged to Judge Hanen that it was siphoning money and manpower from other activities to prepare for the millions of applications, documents, fingerprint checks and other duties required to carry out the amnesty.

One public interest law firm has asked for an investigation to determine whether the planning violated the Antideficiency Act, which makes it illegal to spend money unless Congress has explicitly authorized it.

Ironically, USCIS is the one agency within Homeland Security that will keep operating and paying its employees if the department’s funding lapses. As a fee-based agency, its money is continuing.

But 30,000 other department employees face furloughs after Friday, and tens of thousands more who are deemed “essential” face the prospect of reporting to work without pay until the impasse is resolved.

“The clock is ticking,” said Homeland Security Secretary Jeh Johnson, who held a press conference to highlight other dangers of a funding lapse. He said line agents would remain on the job but most of his staff at headquarters would be furloughed. Those are the ones who organize the agents to “stay one step ahead” of terrorist threats, aviation security problems and illegal immigration trends.

Federal Emergency Management Agency chief Craig Fugate said he would have to stop funding to rebuild from disasters such as Hurricane Katrina and Superstorm Sandy, and the floods that devastated Colorado in 2013.

Mr. Kerlikowske said he would have to close training classes for agents and officers, send the candidates home and hope they don’t find other jobs during the shutdown.

Senate Majority Leader Mitch McConnell, Kentucky Republican, announced a stand-alone bill late Monday that would cancel the latest amnesty while leaving the 2012 version for Dreamers intact. He said it would put Democrats on the spot.

“It isn’t tied to DHS funding. It removes their excuse,” Mr. McConnell said.

His move paves the way for Republicans to pass a short-term bill to get the Homeland Security Department over a Friday funding deadline while preserving the fight over Mr. Obama’s executive actions.

House Speaker John A. Boehner, Ohio Republican, praised Mr. McConnell’s action.

• Dave Boyer contributed to this report.

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