Quartet impressed by dashing flirt (9)
I believe the answer is:
gallivant
'flirt' is the definition.
(thesaurus)
'quartet impressed by dashing' is the wordplay.
'quartet' becomes 'IV' (Roman numerals for four).
'impressed by' is an insertion indicator.
'dashing' becomes 'gallant' (synonyms).
'iv' put within 'gallant' is 'GALLIVANT'.
(Other definitions for gallivant that I've seen before include "Pleasure-seek" , "play around" , "Gad about, seeking pleasure" , "go in search of pleasure" , "run around" .)