Cool Mrs Graham runs educational establishment (7,6)
I believe the answer is:
grammar school
'educational establishment' is the definition.
(I've seen this before)
'cool mrs graham runs' is the wordplay.
'runs' is an anagram indicator.
'cool'+'mrs'+'graham'='coolmrsgraham'
'coolmrsgraham' with letters rearranged gives 'GRAMMAR SCHOOL'.
(Other definitions for grammar school that I've seen before include "Educational establishment" , "Teaching establishment" .)