Irritates the French following row (7)
I believe the answer is:
rankles
'irritates' is the definition.
(rankling is a kind of irritating)
'the french following row' is the wordplay.
'the french' becomes 'les' (les is the in french).
'following' says to put letters next to each other.
'row' becomes 'rank' (I've seen this before**).
'les' put after 'rank' is 'RANKLES'.
(Other definitions for rankles that I've seen before include "Causes lasting and bitter annoyance" , "Continues to hurt" , "Goes on vexing" , "Riles" .)