WASHINGTON — The certificate of appreciation that Hanadi Al Haidari’s family received for providing shelter, food and translation services for the United States military still looks brand-new, without even a crease. She keeps it next to her Iraqi passport in her new home in Denver.

The document is both proof of the risk the family took to assist American soldiers and a reminder of a promise unkept. Ms. Al Haidari’s brother, Ahmed, whose work as a translator for the troops allowed his family to apply for a priority refugee visa to the United States, remains in limbo in the Middle East, struggling to support his 9-year-old son.

“He just wants the basic rights of a normal person,” Ms. Al Haidari said, adding that she did not blame any specific official or government for the delay in approval for her brother’s resettlement. But she was also quick to note that her family’s displacement was rooted in the U.S. invasion of Iraq and the ensuing upheaval. “We wanted to come here because we don’t have a home,” she said. “We don’t have a country anymore.”

The Trump administration had reserved 4,000 slots for Iraqi refugees who had helped American troops, contractors or news media or who are members of a persecuted minority group in the fiscal year that ended Sept. 30. It ultimately admitted only 161 Iraqis — or 4 percent — to the United States, the lowest percentage of the four categories of refugees the administration authorized for resettlement last year. While the coronavirus pandemic caused refugee flights to be canceled for months, immigration lawyers also cited the lasting effects of President Trump’s initial refugee bans and expanded vetting of those fleeing persecution. Of the 5,000 slots reserved for victims of religious persecution, 4,859 were filled — a reflection, perhaps, of the administration’s political priorities.

Ms. Al Haidari’s hopes for her family’s reunion dimmed further last month when Mr. Trump told Congress he planned to cut the cap on refugees for a fourth straight year. The number of refugees admitted depends on the administration and world events, but the ceiling for the current fiscal year, 15,000, is the lowest in the program’s four decade-long history. During the Obama administration, the cap was at least 70,000 a year. The announcement came as Mr. Trump fell back on the kind of anti-immigration messaging that has been a staple of his campaigns, tarring refugees as threats to public safety and the economy, despite multiple studies debunking such generalizations. He also used the issue to attack his Democratic opponent, former Vice President Joseph R. Biden Jr., who has proposed raising refugee admissions to 125,000.

But families like the Al Haidaris make for unlikely political targets. Veterans and active-duty service members fear that the exclusion of those who assisted the military from resettlement is the real threat to national security because such cooperation will be harder to come by in future conflicts. More than 9,800 Iraqis were welcomed to the United States in 2016, according to State Department data. By the 2019 fiscal year, that was down to 465.

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