Flesh is cut -- that's mean (7)
I believe the answer is:
selfish
'mean' is the definition.
(thesaurus)
'flesh is cut' is the wordplay.
'cut' is an anagram indicator.
'flesh'+'is'='fleshis'
'fleshis' is an anagram of 'SELFISH'.
'that's' acts as a link.
(Other definitions for selfish that I've seen before include "Concerned chiefly with one's own interests" , "for personal gain" , "without consideration" , "Concerned only with one's personal welfare" , "Concerned only with oneself" .)