Irritate having the French following status (6)
I believe the answer is:
rankle
'irritate' is the definition.
(I know that rankle has the is a more specific form of the action irritate)
'the french following status' is the wordplay.
'the french' becomes 'le' ('the' in French).
'following' is a charade indicator (letters next to each other).
'status' becomes 'rank' (rank is a kind of status).
'le' after 'rank' is 'RANKLE'.
'having' acts as a link.
(Other definitions for rankle that I've seen before include "Continue to vex" , "Continue to annoy" , "Irritate for a long time" , "Eat into and cause resentment" , "Cause lasting bitterness and resentment" .)