When Representative Juan Ciscomani appeared onstage at a community center filled with banquet tables in Tucson recently to speak to high school seniors from underserved communities, he recounted his personal story of immigrating from Mexico as a young child, helping to support his family and eventually making it to Congress.

As he spoke about translating for his parents during school conferences and weekends spent washing cars to earn extra money to help support his family, Mr. Ciscomani made few mentions of the fact that he is a Republican — and no mention of his party’s presidential nominee, Donald J. Trump, who has made a political career of demonizing immigrants.

“I became a U.S. citizen in 2006 — 13 years after coming to this country,” Mr. Ciscomani told the crowd of community leaders interspersed with families of the young graduates, many of them first-generation Americans from some of the poorest parts of the city. “And then a member of the United States Congress 16 years later. So I believe in the American dream. I believe in the opportunities this country offers.”

It is a pitch that Mr. Ciscomani, 41, a first-term congressman, has leaned on since launching his first campaign for Congress in 2021 in a district that President Biden won the previous year, and is using again as he seeks re-election in one of the most competitive House races in the country.

The task has grown more complicated now that Mr. Ciscomani has a record to defend, including voting for an immigration crackdown measure that Republicans pushed through the House last year, which stalled in the Senate, and opposing a bipartisan bill that would have imposed tough new border enforcement policies and steered billions to funding them.