The first 10 days of 2024 have pulled back the curtain on the core incentive animating the House GOP's election-year agenda: Don't do anything that could potentially rescue President Biden's dreadful approval rating.

Why it matters: Every opposition party strives to be a thorn in the side of the administration in power. But Trump-allied Republicans in Congress appear to be taking their role a step further, stifling any hint of bipartisan compromise while actively pushing to impeach Biden and his top officials.

Driving the news: One of Biden's top vulnerabilities is the migrant crisis, an issue that has overwhelmed cities across the country and spurred substantive bipartisan negotiations in the Senate.

• But while Senate Republicans appear to be working in good faith on a deal to impose new border restrictions, House Republicans are signaling they have no intention of bailing Biden out on his biggest liability.
• "I will not help the Democrats try to improve this man's dismal approval ratings. I'm not going to do it. Why would I?" Rep. Troy Nehls (R-Texas) told CNN last week.

Zoom in: House Speaker Mike Johnson (R-La.) has vowed not to support any border compromise that doesn't contain the GOP's hardline H.R. 2 bill, which would codify some of former President Trump's strictest border policies.

• Johnson told radio host Hugh Hewitt on Wednesday that Republicans would "die on that hill" — despite being well aware the legislation could never pass the Democratic-controlled Senate.
• The speaker also he planned to speak with Trump on Wednesday and predicted Republicans would "be in a total different ballgame next January" after winning the House and Senate.

In the meantime, House Republicans are plowing ahead on their efforts to impeach both Biden and Homeland Security Secretary Alejandro Mayorkas.

• House Oversight Committee Chair James Comer (R-Ky.) suggested this week that Attorney General Merrick Garland also could be targeted with impeachment if Hunter Biden isn't arrested on contempt charges.
• And Rep. Matt Rosendale (R-Mont.) introduced articles of impeachment against Secretary of Defense Lloyd Austin on Monday for waiting several days to inform the White House of his hospitalization.

Between the lines: Comments Trump made this week crystalize the extremes to which some Republicans appear willing to go to damage Biden ahead of the election.

• The GOP front-runner predicted in an interview Monday that the economy will crash — and said he hopes it happens on Biden's watch.
• "When there's a crash, I hope it's going to be during this next 12 months because I don't want to be Herbert Hoover," Trump said, referring to the president who oversaw the onset of the Great Depression.

What they're saying: "Donald Trump should just say he doesn't give a damn about people, because that's exactly what he's telling the American people when he says he hopes the economy crashes," Biden campaign manager Julie Chavez-Rodriguez said in a statement.

What to watch: Negotiations over the child tax credit (CTC) will be a key litmus test for whether House Republicans are willing to grant Biden any semblance of a legislative victory this year.

• House and Senate negotiators are inching closer to a deal to resurrect the CTC in exchange for restoring popular business tax benefits, but GOP leadership's lack of involvement in the talks could present a huge obstacle.
• "I haven't even had a chance to look at it yet," Johnson told CNBC when asked about the potential deal today.