House Republicans will try on Tuesday for a second time to impeach Alejandro N. Mayorkas, the homeland security secretary, on charges of willfully refusing to enforce border laws and breaching the public trust, after their first attempt at the partisan indictment ended in a stunning defeat.

Three Republicans joined all Democrats last week in rejecting the impeachment charges, leaving the G.O.P., which has a tiny margin, just one vote short of a majority in a humiliating spectacle on the House floor.

The decisive moment came when Representative Al Green, Democrat of Texas, who Republicans had counted on missing the vote, arrived in a hospital garb fresh out of abdominal surgery to cast his “no” vote. With Representative Steve Scalise, Republican of Louisiana and the majority leader, absent while he underwent treatment for blood cancer, the G.O.P. was unable to make up the shortfall.

Republicans called Mr. Scalise back to Washington this week, and they were confident on Tuesday that their second attempt would be successful. That would put Mr. Mayorkas in the company of past presidents and administration officials who have been impeached on allegations of personal corruption, election interference and even fomenting an insurrection.

But the charges Republicans have levied have broken with history by failing to identify any such offense, instead effectively declaring the policy choices of the Biden administration that he has carried out a constitutional crime. The approach threatened to lower the bar for impeachments — which already has fallen in recent years — reducing what was once Congress’s most powerful tool to remove despots from power to a weapon to be deployed in political fights.