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