Frenchman in row starts next to Barney in the local (8)
I believe the answer is:
Here is my best explanation:
'barney in the local' is the definition.
'argument' can be an answer for 'barney' (thesaurus). I am not sure about the 'in the local' bit.
'frenchman in row starts next to' is the subsidiary indication.
'frenchman' becomes 'm' (abbreviation for Monsieur - like 'Mr').
'in' is an insertion indicator.
'row' becomes 'argue' (I've seen this before**).
'starts' indicates taking the first letters.
The initial letters of 'next to' is 'nt'.
'm' inserted inside 'argue' is 'argume'.
(Other definitions for argument that I've seen before include "Dispute", "Opposed points of view", "Discussion - line of proof", "Clash of opinions", "Heated disagreement", "Barney", "that may be constructed on premises", "Debate - disagreement", "'Debate, row (8)'" and "Contentious debate".)