The most significant change Mr. de Blasio proposed was replacing the test, called the SHSAT, with a new method that would admit students based on their class rank at their middle school and their scores on statewide standardized tests. That change would require approval from the State Legislature, which has shown little appetite for such a move. A bill outlining those changes was introduced in the Assembly on Friday.

Mr. de Blasio announced another, smaller change on Saturday, one the city can do on its own. Beginning in the fall of 2019, the city would set aside 20 percent of seats in each specialized school for low-income students who score just below the cutoff; those students would be able to earn their spot by attending a summer session called the Discovery program. Five percent of seats for this year’s ninth graders were awarded this way, the city said.

A spokesman for the city’s Education Department said the way students were chosen for the Discovery program would also change. Currently, poor students with certain scores from all over the city qualify, but under the new plan, the city would target students from high-poverty schools instead. Those schools tend to have a higher proportion of black or Hispanic students.