Conclude review you shouldn't have written about private war (8)
I believe the answer is:
vendetta
'private war' is the definition.
(I know that private war can be written as vendetta)
'conclude review you shouldn't have written about' is the wordplay.
'conclude' becomes 'end' (I've seen this before).
'review' becomes 'vet' (I've seen this in other clues).
'you shouldn't have' becomes 'ta' (both can mean 'thank you').
'written about' is an insertion indicator.
'vet'+'ta'='vetta'
'end' inserted within 'vetta' is 'VENDETTA'.
(Other definitions for vendetta that I've seen before include "Lasting feud associated with Sicily and mafia" , "A prolonged feud, associated with Sicily" , "ding-dong battle" , "Blood-feud between families" , "A prolonged feud of Sicilian origin" .)