Sen. John Liu, a Democratic state senator who represents a large Asian population in Queens, helped ensure that the specialized school plan failed in Albany this past summer. “It’s important to be a chancellor for all school kids,” Mr. Liu said. “Two years in, Carranza is falling behind on the curve on that.”

The chancellor has also faced criticism from Representative Grace Meng, a Queens Democrat, along with other Asian-American Democrats in the New York City Council and State Legislature.

And the chasm continues to deepen.

Since the start of this school year, Asian-American parents have accused Mr. Carranza of prejudice over a range of issues, from his decision not to fire an elected parent leader who called Asians “yellow folks” to the fact that the city considered relocating one of the chancellor’s upcoming town hall meetings from Chinatown to Manhattan’s Chelsea neighborhood, before reversing course.

The vitriol and racial division that defined the battle over the so-called specialized high schools have now begun to color a series of seemingly unrelated issues — and stand to influence the school district for years to come.

Shortly after Mr. Carranza arrived in New York in spring 2018, he and Mayor Bill de Blasio announced a plan to get rid of the admissions exam for eight specialized schools, in an effort to boost the number of black and Hispanic students.

Asian students make up a majority of the schools and would lose about half their seats under the city’s proposal. Asian-American parents and politicians were not consulted about the plan before it was announced.