Father has hesitation getting cat (4)
I believe the answer is:
puma
'cat' is the definition.
(puma is a kind of cat)
'father has hesitation' is the wordplay.
'father' becomes 'pa' (pa is a kind of father**).
'has' is an insertion indicator.
'hesitation' becomes 'um'.
'pa' enclosing '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" .)