In the welfare office of the District Department of Human Services, or DHS, in Southeast Washington, Supervisor Rene Smith is conducting an intake interview with a woman in her 20s. The woman’s 3-year-old daughter fidgets on her lap, while a desk fan blows behind Smith, stirring the humid air.

“I’m looking for anything to get my foot in the door,” said Ms. Johnson, who declined to have her first name used. “I’m a fast learner, so I’ll take anything around.”