Cat turned up with old woman (4)
I believe the answer is:
puma
'cat' is the definition.
(puma is a kind of cat)
'turned up with old woman' is the wordplay.
'turned' says the letters should be written in reverse.
'with' is a charade indicator (letters next to each other).
'old woman' becomes 'ma' (both can mean someone's mother).
'up' back-to-front is 'pu'.
'pu'+'ma'='PUMA'
(Other definitions for puma that I've seen before include "Mountain lion" , "Large mountain cat" , "Large wild American feline" , "South American cat" , "Large Am. wild cat" .)