A letter to study hard, then buzz off (5)
I believe the answer is:
scram
'buzz off' is the definition.
('scram' can be a synonym of 'buzz off')
'a letter to study' is the wordplay.
'a letter' becomes 's' (**).
'to study' becomes 'cram' (cramming is a kind of studying).
's'+'cram'='SCRAM'
'hard then' acts as a link.
This may not be right. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for scram that I've seen before include "Go away quickly" , "Scarper" , "Buzz off!" , "Clear off!" , "Skedaddle!" .)