Top state lawmakers called for new efforts to change admissions for New York City’s eight specialized high schools, after data this week showed less than 11% of black and Latino students were offered seats there for next fall.

Seven black students got into the most selective, Stuyvesant High School, down from 10 a year ago. Thirty-three Latino students got offers there, out of 895 eighth-graders admitted through the single exam that determines access.

The...