Cat turning up with old lady (4)
I believe the answer is:
puma
'cat' is the definition.
(I know that puma is a type of wildcat)
'turning up with old lady' is the wordplay.
'turning' shows that the letters should be reversed in order.
'with' says to put letters next to each other.
'old lady' becomes 'ma' (both can mean mother).
'up' written backwards gives 'pu'.
'pu'+'ma'='PUMA'
(Other definitions for puma that I've seen before include "Cougar" , "American wild cat" , "South American carnivore" , "Large Am. wild cat" , "International rugby player" .)