Father, with hesitation, gets cat (4)
I believe the answer is:
puma
'cat' is the definition.
(puma is a kind of cat)
'father with hesitation' is the wordplay.
'father' becomes 'pa' (pa can mean dad or father).
'with' indicates putting letters inside.
'hesitation' becomes 'um' (sound made when hesitating).
'pa' enclosing 'um' is 'PUMA'.
'gets' acts as a link.
(Other definitions for puma that I've seen before include "Mountain lion" , "South American carnivore" , "US wild cat" , "South American cat" , "back from Argentina" .)