Offensive racket about old Frenchman (7)
I believe the answer is:
noisome
'offensive' is the definition.
(I know that offensive can be written as noisome)
'racket about old frenchman' is the wordplay.
'racket' becomes 'noise' (racket can mean a loud noise).
'about' is an insertion indicator.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'frenchman' becomes 'M' (abbreviation for Monsieur - like 'Mr').
'o'+'m'='om'
'noise' placed around 'om' is 'NOISOME'.
(Other definitions for noisome that I've seen before include "Having an extremely offensive smell" , "Nauseating" , "Obnoxious" , "Repellent" , "Disgusting to sight or smell" .)