Offensive racket around the heart of home (7)
I believe the answer is:
noisome
'offensive' is the definition.
('noisome' can be similar in meaning to 'offensive')
'racket around the heart of home' is the wordplay.
'racket' becomes 'noise' ('noise' can be a synonym of 'racket'**).
'around' indicates putting letters inside.
'the heart of' indicates the central letters.
The middle letters of 'home' are 'om'.
'noise' enclosing 'om' is 'NOISOME'.
(Other definitions for noisome that I've seen before include "Obnoxious" , "injurious to health" , "Disagreeable - noxious" , "Smelling bad" , "terribly unpleasant" .)