Fighter catches Greek priestess dressing (5)
I believe the answer is:
aioli
'dressing' is the definition.
(I've seen this before)
'fighter catches greek priestess' is the wordplay.
'fighter' becomes 'Ali' (boxer Muhammad Ali).
'catches' is an insertion indicator.
'greek priestess' becomes 'Io' (priestess in Greek myth).
'ali' placed around 'io' is 'AIOLI'.
(Other definitions for aioli that I've seen before include "garlic sauce" , "Garnish" , "Mayo, in part?" , "Sort of mayonnaise" , "Pungent sauce" .)