Teaching establishment getting rid of coach (9,6)
I believe the answer is:
finishing school
'teaching establishment' is the definition.
(school for upper-class girls)
'getting rid of coach' is the wordplay.
'getting rid of' becomes 'finishing' (both can mean killing).
'coach' becomes 'school' (both can mean to teach or train).
'finishing'+'school'='FINISHING-SCHOOL'
(Other definitions for finishing school that I've seen before include "breaking up?" , "that trains young ladies" , "Private college for girls" , "Educational establishment" .)