Father has hesitation getting cat (4)
I believe the answer is:
puma
'cat' is the definition.
(I know that puma is a type of wildcat)
'father has hesitation' is the wordplay.
'father' becomes 'pa' (pa can mean dad or father).
'has' is an insertion indicator.
'hesitation' becomes 'um' (um is said in hesitation).
'pa' going around 'um' is 'PUMA'.
'getting' is the link.
(Other definitions for puma that I've seen before include "back from Argentina" , "Large American feline resembling a lion" , "Panther" , "South American carnivore" , "American wild cat" .)