Father, with hesitation, gets a cat (4)
I believe the answer is:
puma
'a 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' means one lot of letters goes inside another.
'hesitation' becomes 'um' (sound made when hesitating).
'pa' going around 'um' is 'PUMA'.
'gets' acts as a link.
(Other definitions for puma that I've seen before include "Panther" , "Cougar" , "US wild cat" , "South American cat" , "American wildcat" .)