Close family member the French rejected after a short time (9)
I believe the answer is:
secretive
'after a short time' is the definition.
I can't tell whether this definition defines the answer.
'close family member the french rejected' is the wordplay.
'close' becomes 'sec' (I am not sure about this - if you are sure you should believe this answer much more).
'family member' becomes 'relative' (I've seen this before).
'the french' becomes 'la' ('the' in French).
'rejected' suggests deleting specific letters.
'relative' with 'la' taken out is 'retive'.
'sec'+'retive'='SECRETIVE'
(Other definitions for secretive that I've seen before include "Inclined to concealment" , "Unforthcoming" , "giving nothing away" , "Very private" , "withdrawn" .)