Coax salesman over petition to limit publicity (8)
I believe the answer is:
persuade
'coax' is the definition.
(I know that coax is a more specific form of persuade)
'salesman over petition to limit publicity' is the wordplay.
'salesman' becomes 'rep' (sales representative).
'over' says the letters should be written backwards.
'petition' becomes 'sue' (both can mean to appeal or ask for something).
'to limit' is an insertion indicator (outside letters limit or restrain others).
'publicity' becomes 'ad' (short for advert).
'rep' back-to-front is 'per'.
'sue' enclosing 'ad' is 'suade'.
'per'+'suade'='PERSUADE'
(Other definitions for persuade that I've seen before include "influence" , "Convince by reasoning" , "Convince by argument" , "Urge successfully" , "Talk round" .)