Businessmen make Carol sigh sadly (9)
I believe the answer is:
oligarchs
'businessmen' is the definition.
(I have seen 'Influential businessmen' mean 'oligarchs' so perhaps 'businessmen' could also mean 'oligarchs')
'carol sigh sadly' is the wordplay.
'sadly' indicates anagramming the letters.
'carol'+'sigh'='carolsigh'
'carolsigh' anagrammed gives 'OLIGARCHS'.
'make' acts as a link.
(Other definitions for oligarchs that I've seen before include "Influential businessmen" , "governors" .)