To annoy the French with military status (6)
I believe the answer is:
rankle
'to annoy' is the definition.
(rankling is a kind of annoying)
'the french with military status' is the wordplay.
'the french' becomes 'le' ('the' in French).
'with' means one lot of letters go next to another.
'military status' becomes 'rank' (I've seen this in another clue).
'le' after 'rank' is 'RANKLE'.
(Other definitions for rankle that I've seen before include "Cause lasting annoyance or resentment" , "Irritate for a long time" , "Cause lasting bitterness and resentment" , "Cause persistent annoyance" , "Continue vexing" .)