Month in dispute — something black about that old conflict (6,3)
I believe the answer is:
trojan war
'old conflict' is the definition.
(war in Greek mythology)
'month in dispute something black about that' is the wordplay.
'month' becomes 'Jan' (Jan is short for January).
'in' is an insertion indicator.
'dispute' becomes 'row' (row is a kind of dispute).
'something black' becomes 'tar' (tar is a black substance).
'about that' means one lot of letters goes inside another.
'jan' inserted inside 'row' is 'rojanw'.
'rojanw' placed into 'tar' is 'TROJAN WAR'.
(Other definitions for trojan war that I've seen before include "Long battle" , "Ancient conflict" , "protracted conflict" , "Conflict recorded by Homer" , "dispute of old" .)