Small minded institute removes qualified teacher (4)
I believe the answer is:
mini
'small' is the definition.
(I've seen this before)
'minded institute removes qualified teacher' is the wordplay.
'institute' becomes 'i' (common abbreviation as in MIT).
'removes' suggests deleting specific letters.
'qualified teacher' becomes 'ded' (abbreviation for Diploma in Education).
'minded'+'i'='mindedi'
'mindedi' with 'ded' taken out is 'MINI'.
(Other definitions for mini that I've seen before include "Baby" , "Italian Job car?" , "Short skirt - small car" , "Cult British car" , "Brief" .)