Annoyed, it's clear, the French start to depart (7)
I believe the answer is:
nettled
'annoyed' is the definition.
(I know that annoyed can be written as nettled)
'clear the french start to depart' is the wordplay.
'clear' becomes 'nett' (similar in meaning).
'the french' becomes 'le' ('the' in French).
'start to' suggests taking the first letters.
The first letter of 'depart' is 'd'.
'nett'+'le'+'d'='NETTLED'
'it's' is the link.
(Other definitions for nettled that I've seen before include "Annoyed, irritated" , "Provoked" , "Riled" , "becomes annoyed" , "annoyed one" .)