WASHINGTON — Last week, the Republican Party suffered a stinging blow due to its narrow majority when a resolution to impeach Homeland Security Secretary Alejandro Mayorkas fell short by a mere one vote and left party leaders stunned.

It marked an embarrassing defeat for Speaker Mike Johnson, R-La., and caused a wave of recriminations as Republicans lament a thin 219-to-212 majority that has left them unable to govern — and brought nonstop mayhem across Capitol Hill.

Next week, that margin will change after voters in New York’s 3rd Congressional District, which spans portions of Queens and Long Island, have their say. The only question is whether the Republicans’ majority will shrink even further, or if they’ll regain a vote they lost after the expulsion of Rep. George Santos, R-N.Y., who was ousted after a 23-count federal indictment alleging a smorgasbord of crimes.

The contest pits former Democratic Rep. Tom Suozzi, who is running as an independent-minded legislator willing to break with his party, against Republican Nassau County Legislator Mazi Pilip, who is seeking to ride discontent over migration and crime to a win. It’s a close race: A Newsday/Siena College survey out Thursday found Suozzi leading by 4 points, inside the margin of error.

The outcome will affect Republican hopes of impeaching other officials, including President Joe Biden, as well as must-pass legislation like funding the government, reauthorizing the Federal Aviation Administration, approving a farm bill and deciding whether to renew warrantless surveillance powers.

If the last week has taught members of Congress one thing, it’s that one vote can make all the difference.

House GOP leaders are eyeing another vote Tuesday to impeach Mayorkas, although it would likely require perfect attendance from the majority party and the expected return of Majority Leader Steve Scalise, R-La., who has been on medical leave battling cancer. If the House falls short again on Tuesday, the outcome in New York could determine whether Republicans have the votes to impeach him.

The two candidates are split on the issue along party lines.

“I will vote him out. I will impeach him,” Pilip said in an interview. “I think he failed to protect the voters. He failed doing his job.”

Suozzi called the impeachment attempt “a political witch hunt” that won’t fix the border crisis, arguing that only legislation can.

“As some Republicans have said, it doesn’t even make sense to do this,” he said in a recent forum hosted by PIX11. “You want to fix the border? You’re not going to fix it by impeaching the secretary. You’re gonna fix it by doing bipartisan legislation to finally fix it.”

The wafer-thin House majority has made it difficult for Republican leaders to hold votes at all, as a small group of rebels can block them. Instead, Johnson has regularly been forced to use “suspension of the rules” to hold votes, a process that requires a two-thirds majority to pass but bypasses a procedural vote his detractors have been using to kill legislation.

The failed Mayorkas vote left some in the House GOP wondering whether it was a mistake to expel Santos. Others lamented a couple of recent resignations that thinned their margins, including that of former Rep. Kevin McCarthy, R-Calif., the ousted speaker.

Rep. Tim Burchett, R-Tenn., lamented that the GOP majority is so small that Democrats were able to sink the Mayorkas impeachment by bringing Rep. Al Green, D-Texas, back from the hospital to vote in a wheelchair.

“And of course, Steve Scalise is out with cancer. … After our former speaker left us and became a lobbyist, it kind of left us a little bit in a lurch,” Burchett, who voted to oust McCarthy, said. “So, I think it’ll be brought back up and I think I think it’ll be successful.” (McCarthy hasn’t announced his new job yet since leaving Congress.)

It’s swing districts like this one, which voted for Biden and Suozzi in 2020, then flipped Republican in 2022, that will decide whether House Minority Leader Hakeem Jeffries, D-N.Y., will remain minority leader or become speaker next year.

“The special election on Tuesday is a race between Tom Suozzi and the Republican candidate, who has embraced the most extreme positions on the other side of the aisle,” said Jeffries. “Tom Suozzi is a commonsense, get-stuff-done kind of public servant.”

“And let’s see what happens in the aftermath of the election as to what it all means,” he said.

Last week, Johnson paid a visit to New York to help Pilip.

“I’m here in New York because Mazi Pilip,” he said in a Fox Business interview on Feb. 9. “She’s going to get that job done. It’ll send a strong message to the Democrats that coming into this election cycle we’ve got a great run, great candidates, we’re feeling really bullish about the future.”