Turning up with mice primarily, like cats (5)
I believe the answer is:
pumas
'cats' is the definition.
(puma is a kind of cat)
'turning up with mice primarily like' is the wordplay.
'turning' says the letters should be written backwards.
'with' is a charade indicator (letters next to each other).
'primarily' suggests taking the first letters.
'like' becomes 'as'.
The initial letter of 'mice' is 'm'.
'up' backwards is 'pu'.
'pu'+'m'+'as'='PUMAS'
(Other definitions for pumas that I've seen before include "North American mountain lions" , "Large American felines" , "Big cats" , "American wild cats" , "Big felines" .)