“The client is grieving the possibility of not seeing the therapist again,” Ms. Siu said. “So saying goodbye with hugs and tears each time is a form of control, because it’s on their own terms.”

El Futuro has waiting lists of people who want help. But in a survey of patients, the clinic found that some people are afraid to come in. Elizabeth, 27 and here illegally, is among them.

With great reluctance she showed up at the clinic for an interview with a reporter, arriving late, uneasy. Apologizing, she said she leaves her apartment these days only to go to the grocery store and to her job as a hotel maid.

She has no one who will care for her two young children if she is deported, she explained haltingly, tears welling up.

And in Mexico, another danger awaits: her ex-boyfriend. Years ago, when the couple arrived in North Carolina, she said, he began to beat her so badly that she finally called the police. They arrested him and had him deported.

Now, fearful for her children and for her own safety, Elizabeth is consumed by anxiety. Her nightmares from that violent period are back.

After recounting her story, Elizabeth walked toward El Futuro’s reception area, clutching her 5-year-old daughter’s hand. Even if she cleared the clinic’s wait list, she said, it just seemed too risky to come back.