A quiet moment spare in museum (9)
I believe the answer is:
ashmolean
'museum' is the definition.
(I've seen this before)
'a quiet moment spare' is the wordplay.
'quiet' becomes 'sh'.
'moment' becomes 'mo' (slang abbreviation. e.g. 'just a mo').
'spare' becomes 'lean' (similar in meaning).
'a'+'sh'+'mo'+'lean'='ASHMOLEAN'
'in' acts as a link.
(Other definitions for ashmolean that I've seen before include "Oxford institution" , "Oxford museum" .)