RHINEBECK, N.Y. — On a recent Friday, as impeachment investigators were releasing new evidence of President Trump’s pressure campaign on Ukraine and John R. Bolton, the former national security adviser, was refusing through his lawyer to tell Congress what he knew about “many relevant meetings and conversations” in the matter, Representative Antonio Delgado’s mind was elsewhere.

Driving through his Hudson Valley district under a light snow, Mr. Delgado, a first-term Democrat, was preoccupied with legislation he was pursuing, toxic chemical cleanup provisions he was fighting to include in a must-pass defense bill, the farmers he was scheduled to meet and his push to hold 33 town hall-style meetings by the end of December.

Before the sun set at 4:45 p.m., Mr. Delgado would trek 175 miles across the district, scarfing down a breakfast of fluffy pancakes with his wife and twin sons early in the morning so he could attend a school assembly, then making last-minute edits to a speech in the car in between events. It barely left time to brush up on the latest news in the impeachment inquiry consuming Washington.

With five legislative days remaining before Congress leaves for its holiday break, the House is charging toward a party-line vote on impeachment articles against Mr. Trump that will make him only the third president to be impeached. The vote is a politically risky one for freshman lawmakers like Mr. Delgado who won by narrow margins in districts that supported Mr. Trump in 2016, and one that the president and his political allies have argued will cost them their seats.