With his plans to integrate New York City’s eight specialized high schools, Mayor Bill de Blasio is pushing for fundamental changes to address a lack of diversity at these elite institutions, which admit students by a single exam score.

In one definite step, he said 20% of ninth-grade slots will be set aside for low-income students who just miss the test-score cutoff for entry to these schools, which include Stuyvesant High School, starting in fall 2019.

In...