KINSMAN, Ohio — For years, Tammy and Joseph Pavlic tried to ignore the cracked ceiling in their living room, the growing hole next to their shower and the deteriorating roof they feared might one day give out. Mr. Pavlic worked for decades installing and repairing air-conditioning and heating units, but three years ago, with multiple sclerosis advancing, he had to leave his job.

By 2015, Ms. Pavlic was supporting her husband and their three children on an annual salary of $9,000, earned at a restaurant. That year, they tapped a county program funded by Congress, called the HOME Investment Partnerships Program, to help repair their house.

The next year, they voted for Donald J. Trump, who has moved to eliminate the HOME program.

The Pavlics’ ceiling may no longer be cracked, but in the zero-sum game that Mr. Trump’s budget seeks to set up, the nation is showing new fissures. The president’s budget proposal would cut deeply into the Department of Housing and Urban Development, paring rental assistance and eliminating heating and air-conditioning aid, energy-efficiency assistance, and partnerships with local governments like HOME. With the savings, Mr. Trump says, he would beef up military spending and build a wall along the Mexican border.

“Keeping the country safe compared to keeping my bathroom safe isn’t even a comparison,” Mr. Pavlic, 42, said. “We have people who are coming into this country who are trying to hurt us, and I think that we need to be protected.”