Backing school, the French finally sent brief communication (7)
I believe the answer is:
notelet
'brief communication' is the definition.
(I've seen this before)
'backing school the french finally sent' is the wordplay.
'backing school' becomes 'note' (I can't justify this - if you can you should believe this answer much more).
'the french' becomes 'le' ('the' in French).
'finally' indicates one should take the final letters.
The last letter of 'sent' is 't'.
'note'+'le'+'t'='NOTELET'
Can you help me to learn more?
(Other definitions for notelet that I've seen before include "Paper folded" , "correspondence" , "informal communication" , "Small written communication" , "Small sheet for letter" .)