Sarita Latchman, a vibrant 42-year-old mother and former parks worker, has a sound like a baby’s rattle at the back of her throat. Which is not surprising, as her apartment in the Jefferson Houses in East Harlem is speckled with soot-black mold. A thick carpet of it runs down her bathroom wall and across the ceiling of her children’s bedrooms. Rub it and the spores float, landing on sink tops and children’s hair. They also journey through Ms. Latchman’s nasal passageway into her lungs.