Month up and exhausted initially, give up (4)
I believe the answer is:
cede
'give up' is the definition.
('cede' can be a synonym of 'give up')
'month up and exhausted initially' is the wordplay.
'month' becomes 'dec' (Dec is an example**).
'up' says the letters should be written in reverse (in down clue: letters go upwards).
'and' is a charade indicator (letters next to each other).
'initially' suggests taking the first letters.
The initial letter of 'exhausted' is 'e'.
'dec' written backwards gives 'ced'.
'ced'+'e'='CEDE'
(Other definitions for cede that I've seen before include "Give up one's rights" , "Yield, give way" , "Give up (power to another)" , "Allow" , "Give up (eg, power)" .)