Father, at home, is a nuisance (4)
I believe the answer is:
pain
'nuisance' is the definition.
(I know that nuisance can be written as pain)
'father at home' is the wordplay.
'father' becomes 'pa' (pa can mean dad or father).
'at home' becomes 'in' ('I'm in' can mean 'I'm at home').
'pa'+'in'='PAIN'
'is a' is the link.
(Other definitions for pain that I've seen before include "Annoying person" , "Hurt feeling" , "Mental distress" , "Unpleasant sensation" , "nuisance to be put up with" .)