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 is a kind of father**).
'with' indicates putting letters inside.
'hesitation' becomes 'um'.
'pa' going around 'um' is 'PUMA'.
'gets' is the 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" .)