Ask about pursuit (7)
I believe the answer is:
request
'ask' is the definition.
(I know that ask is a more specific form of the action demand)
'about pursuit' is the wordplay.
'about' becomes 're' (regarding).
'pursuit' becomes 'quest' (synonyms).
're'+'quest'='REQUEST'
(Other definitions for request that I've seen before include "Ask for" , "Polite demand" , "Application" , "Petition" , "Queer St (anag.)" .)