Cat turned up with old lady (4)
I believe the answer is:
puma
'cat' is the definition.
(puma is a kind of cat)
'turned up with old lady' is the wordplay.
'turned' shows that the letters should be reversed in order.
'with' means one lot of letters go next to another.
'old lady' becomes 'ma' (both can mean mother).
'up' back-to-front is 'pu'.
'pu'+'ma'='PUMA'
(Other definitions for puma that I've seen before include "Large wild American feline" , "Cougar" , "back from Argentina" , "Large Am. wild cat" , "Panther" .)