Father has hesitation getting a cat (4)
I believe the answer is:
puma
'a cat' is the definition.
(puma is a kind of cat)
'father has hesitation' is the wordplay.
'father' becomes 'pa' (pa can mean dad or father).
'has' means one lot of letters goes inside another.
'hesitation' becomes 'um' (um is said in hesitation).
'pa' enclosing 'um' is 'PUMA'.
'getting' is the link.
(Other definitions for puma that I've seen before include "American wild cat" , "Cougar" , "American wildcat" , "South American cat" , "Large American feline resembling a lion" .)