The French greeting gives permission (5)
I believe the answer is:
leave
'permission' is the definition.
(leave is permission or authorisation)
'the french greeting' is the wordplay.
'the french' becomes 'le' ('the' in French).
'greeting' becomes 'ave' (Latin greeting).
'le'+'ave'='LEAVE'
'gives' acts as a link.
(Other definitions for leave that I've seen before include "Abandon" , "Clearance" , "Don't take - time off" , "Relinquish - holiday" , "ditch" .)