The decision doesn’t say “work to.” It actually passes it back down to the district court to handle the details. Sotomayor wrote the decision:

On Friday, April 4, the United States District Court for the District of Maryland entered an order directing the Government to “facilitate and effectuate the return of [Abrego Garcia] to the United States […] The order properly requires the Government to “facilitate” Abrego Garcia’s release from custody in El Salvador and to ensure that his case is handled as it would have been had he not been improperly sent to El Salvador. The intended scope of the term “effectuate” in the District Court’s order is, however, unclear, and may exceed the District Court’s authority. The District Court should clarify its directive, with due regard for the deference owed to the Executive Branch in the conduct of foreign affairs.

Emphasis mine and citations removed. The decision also said:

The only argument the Government offers in support of its request, that United States courts cannot grant relief once a deportee crosses the border, is plainly wrong. […] The Government’s argument, moreover, implies that it could deport and incarcerate any person, including U. S. citizens, without legal consequence, so long as it does so before a court can intervene. […] That view refutes itself.

Citations removed. And:

I agree with the Court’s order that the proper remedy is to provide Abrego Garcia with all the process to which he would have been entitled had he not been unlawfully removed to El Salvador. That means the Government must comply with its obligation to provide Abrego Garcia with “due process of law,” including notice and an opportunity to be heard, in any future proceedings. […] It must also comply with its obligations under the Convention Against Torture. […] Federal law governing detention and removal of immigrants continues, of course, to be binding as well. […] (requiring a warrant before a noncitizen “may be arrested and detained pending a decision” on removal) […] (in order to revoke conditional release, the Government must provide adequate notice and “promptly” arrange an “initial informal interview . . . to afford the alien an opportunity to respond to the reasons for the revocation stated in the notification”). Moreover, it has been the Government’s own well-established policy to “facilitate [an] alien’s return to the United States if . . . the alien’s presence is necessary for continued administrative removal proceedings” in cases where a noncitizen has been removed pending immigration proceedings.

Emphasis mine and citations removed, again. And finally, she ends it with:

In the proceedings on remand, the District Court should continue to ensure that the Government lives up to its obligations to follow the law.

So I’m not a lawyer, but it looks like her decision is basically saying, “yes, fix it; and District Court, make it airtight and keep an eye on them.” It’s actually pretty forceful, coming from SCOTUS.

“Work to” was invented by the AP to encapsulate all of that.