Hard-right House Republicans agreed late Monday to give their party leaders a temporary reprieve from a weeklong blockade of the House floor, allowing some legislative business to move forward on Tuesday but insisting they would withhold their support for future votes if their demands were not met.

The move counted as progress for Speaker Kevin McCarthy, after days of paralysis in the House that showcased his weak grip on his fractious conference in the face of the rage of right-wing Republicans over the deal he cut with President Biden to suspend the debt limit and avert a federal default. But the agreement was only provisional, and the group of about a dozen ultraconservative lawmakers who have held the floor hostage made it clear they planned to continue using guerrilla tactics to keep a tight leash on Mr. McCarthy, effectively exercising veto power on what he is able to accomplish.

The House remained at a standstill on Monday evening, after Mr. McCarthy canceled scheduled votes. But under a deal he reached with the right-wing members of the House Freedom Caucus, business was set to resume on Tuesday with votes on legislation to guard against restrictions on gas stoves and other government regulations, as well as a separate bill to kill stricter federal rules for pistols with stabilizing braces, according to people familiar with the discussions who spoke about them on the condition of anonymity.

In conversations on Monday, the people said, members of the rebel group were explicit with Mr. McCarthy that he could not count on their support for bringing up any other legislation next week or in the future, until they had worked out a power-sharing agreement that guaranteed them major influence on the legislative agenda.

Last week was the first time in two decades that members of the majority party in the House had sided with the party out of power to defeat a rule, a procedural vote normally considered routine and almost always decided along party lines with the majority unanimously in support. The move meant that, at least temporarily, Mr. McCarthy was left as speaker in name only, with no governing majority behind him. In talks with G.O.P. leadership on Monday, the rebels made it clear that they planned to make a habit of it.

Still, some of the 11 Freedom Caucus members emerged from talks in Mr. McCarthy’s office on Monday saying they would support the rule they helped to defeat last week — to call up the regulatory measures — as long as it was adjusted to include the pro-gun legislation championed by conservative Republicans.

Mr. McCarthy called the meetings “productive” but conceded that more talks would be needed to reach a resolution.

“We know when we work together and work on conservative issues, we were winning, and we get more victories that way,” he told reporters Monday evening at the Capitol. “And I think everybody wants to get back to that place.”

Mr. McCarthy said there appeared to be a “willingness” to work out differences but conceded that “doesn’t mean it is all locked together. It means we thought that meeting was great, so we’re putting other ones together.”

Earlier in the day, Mr. McCarthy said he was willing to keep the House on pause for another few days if it meant Republicans could “get it solved so it doesn’t keep erupting all the time.”

But the resolution reached on Monday did not appear to be a long-term one, and it was not clear what such a solution would look like. Hard-right Republicans are seeking, among other demands, spending caps below what Mr. McCarthy and Mr. Biden agreed to in the debt ceiling bill.

The intraparty feud erupted last week in reaction to the compromise Mr. McCarthy struck with Mr. Biden to suspend the debt limit, which contained only a fraction of the spending cuts Republicans had demanded. The deal enraged ultraconservative Republicans who claim Mr. McCarthy betrayed promises he made to them during his fight for the speakership.

The revolt effectively shut down the House last week, and talks to resolve the impasse dragged into Monday.

The consequences of the rebellion were low stakes for now: The G.O.P. insurgents succeeded in delaying only their own party’s messaging bills, which have no chance of passage in a Democratic-controlled Senate. But it served as a reminder of the difficulty the speaker could have down the line when he has to hold together his conference to pass crucial spending bills, which will be required to avert a government shutdown this fall and punishing spending cuts in early 2025.

It was also reminiscent of the paralysis that gripped the House in January, when some of the same instigators of the current drama refused to back Mr. McCarthy for speaker, prompting a historic 15-round election that highlighted the outsize influence of a small group of right-wing lawmakers in a narrow Republican majority.

Carl Hulse contributed reporting.