Damien, age 5, was giddy with excitement as he left a Manhattan homeless shelter, sometimes running and skipping along the sidewalk accompanied by his wistful mother, a migrant from Ecuador. “What I want for him is a future,” Kimberly Carchipulla said in Spanish of her son, one of nearly 800,000 New York City public school students who headed off to class Thursday for their first day of the new school year. Several major U.S. cities have struggled with an influx of many thousands of asylum seekers who have filled up homeless shelters after entering the U.S.