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Ich denke, die Kinder gehen normalerweise in die "Stammschule" und nur während ihres Krankenhausaufenthaltes sozusagen hilfsweise in die Klinikschule

I think in the U.S. "regular schools" would be a common way of putting it, at least in New York.

I ran this one past a retired German school principal who used to teach English. He says that in cases where a child attends, or a teacher teaches at, more than one school, e.g., public school and clinic school during hospitalization, the Stammschule designates the school in which the child is normally enrolled, or at which the teacher mainly teaches, and would translate it as such. Thus: "With this project, we would especially like to strengthen the cooperation between clinic schools and (local) schools in which the students are normally enrolled...." This approach better clarifies that the students are only temporarily enrolled in the clinic school.