WASHINGTON — Representative Kevin McCarthy of California, the minority leader, on Monday scrounged for the support he would need to become speaker of the House if Republicans gain control of the chamber, facing resistance from a newly emboldened right flank as Republicans grappled with their historically weak performance in the midterm elections.

With the Senate in Democratic hands and control of the House still up in the air, Republicans began the week — which they had expected would be a triumphant victory lap — limping toward the finish and bitterly divided over who should lead what was shaping up to be a tiny and unruly majority.

“No one in this town has 218 votes for speaker of the House, so we’re going to have a debate and make sure that we set up the structure properly to then figure out how someone will get 218 votes,” Representative Chip Roy, Republican of Texas and a member of the ultraconservative Freedom Caucus, said as he emerged from a nearly hourlong meeting in Mr. McCarthy’s Capitol office suite.

The Freedom Caucus has threatened to run a protest candidate against Mr. McCarthy, although no challenger has emerged, and has insisted that he overhaul party rules to weaken the speaker’s power and give more sway to the rank-and-file. If he does not secure the top job, it could make his life extremely difficult. It would allow any Republican to take him out at any time and complicate the task of passing any major legislation through the chamber.

This was one day before McCarthy faced his first test in his bid to lead Congress. On Tuesday, Republicans were to hold internal leadership elections. The vote will be a major barometer for Mr. McCarthy’s standing, though he will still need to win the speakership at the start of the new Congress in an election on the floor in which all members have a vote.

Some right-leaning Republicans made it clear Monday that he was not yet able to win them over.

“I am willing to support anybody that’s willing to change dramatically how things are done here so that members are empowered, fairly and equitably,” The Freedom Caucus chairman, Representative Scott Perry from Pennsylvania, said. He declined to say whether he would back Mr. McCarthy or another cAndidate but said the group’s members had a “cordial” and “productive” Talk to Mr. McCarthy

In a sign of discord within the party over what to do next, some leaders of powerful hard-right groups in Washington wrote Monday requesting that the Senate and House Republicans postpone their leadership elections for this week.

“The Republican Party needs leaders who will confidently and skillfully present a persuasive coherent vision of who we are, what we stand for and what we will do,” The signers included leaders of the Club for Growth (the Heritage Foundation) and the Conservative Partnership Institute. “Many current elections are still undecided. There should be no rushed leadership elections.”

There was no sign that the prodding would have any impact in the Senate. Party leaders have resisted calls to postpone their Wednesday leadership elections. As of Monday, there was no opposition to Senator Mitch McConnell, Kentucky, who was the minority leader. He was expected to keep his position.

More complicated was the calculus for McCarthy. A poor showing on Tuesday in his conference’s internal elections would embolden lawmakers in the party’s right flank, many of whom have sought for years to weaken Mr. McCarthy.

Trump’s presence in the wings made things more complicated. After a series of midterm losses caused by candidates he handpicked, Mr. Trump was expected to announce his intention to run again for president just hours before Mr. McCarthy was due to face him at his conference. This election would also include candidates from other races as well as newly elected Republicans.

“Members have a lot of questions,” Representative Byron Donalds, who is running to be the No. 4 leadership position. “Number one, obviously the expectations were not delivered. Number two, how are we actually going to govern what’s going to be a small majority?”

Mr. Donalds continues: “We have to actually lay out a demonstrated plan; serious policies that voters actually believe in and we can’t equivocate from that, even in what’s looking to be continued gridlock government. Republican members want to get answers to that, and I think that’s going to be critical toward Kevin’s ability to becoming the leader, and eventually the speaker.”

Mr. McCarthy received support on Monday from Representative Marjorie Taylor Greene, Republican of Georgia, one of the party’s most prominent far-right lawmakers who is close to Mr. Trump.

Ms. Greene challenged Mr. McCarthy to a podcast interview. “risky” You can also find out more about a “bad strategy,” It could be used to empower Democrats.

“Here’s the harsh reality,” Ms. Greene spoke. “There are Republicans in our conference that would make a deal with the Democrats and cross over.”