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TURLOCK, Calif. — In all of his campaigns for the House, Representative Jeff Denham has never seen anything quite like the rolling circus that trails him through his sprawling district in California’s Central Valley — the “Dump Denham” signs, the papier-mâché effigies, the shouting.

He says it does not faze him, but there is nothing like the political gallows to focus the mind of an endangered politician, and Mr. Denham, a Republican, is responding in a way that touches almost everybody here in California farm country: He is leading the charge on Capitol Hill to pressure Speaker Paul D. Ryan to hold a vote on legislation to protect the young undocumented immigrants known as Dreamers.

Nearly two dozen other House Republicans — about half of them politically vulnerable and many in districts with large Hispanic populations — have followed suit, affixing their signatures to a petition that is just a few votes short of forcing Mr. Ryan to act. If it works, it will push the party into a divisive, and politically risky, election year debate on immigration.

A vote this summer to help undocumented immigrants could demoralize President Trump’s most ardent supporters and depress Republican turnout in November. A vote to toughen immigration rules and harm the young Dreamers would further energize Democratic voters. To avoid such a showdown, the speaker has scheduled a two-hour meeting on immigration with his rank and file when lawmakers return to Washington next week.