School rotter in A&E? Goodness! (7)
I believe the answer is:
academy
'school' is the definition.
(I know that academy is a type of school)
'rotter in ae? goodness' is the wordplay.
'rotter' becomes 'cad' (I've seen this before).
'in' is an insertion indicator.
'goodness' becomes 'my' (both are exclamations of surprise).
'cad' going inside 'ae' is 'acade'.
'acade'+'my'='ACADEMY'
(Other definitions for academy that I've seen before include "School for specialised education" , "Secondary school" , "Educational institution" , "Place of specialised study or training" , "University" .)