Weary, having been drilled (5)
I believe the answer is:
bored
'drilled' is the definition.
('bore' can be a synonym of 'drill')
'weary having been' is the wordplay.
'weary' becomes 'bore' (as a verb).
'having' means one lot of letters go next to another.
'been' becomes 'd' (I can't explain this - if you can you should give a lot more credence to this answer).
'bore'+'d'='BORED'
Can you help me to learn more?
(Other definitions for bored that I've seen before include "Fed up, afflicted with ennui" , "Drilled; fed up" , "Brassed off" , "Tired and uninterested" , "Weary and impatient" .)