Month up and exhausted initially, give up (4)
I believe the answer is:
Here is my best explanation:
'give up' is the definition.
('cede' can be a synonym of 'give up')
'month up and exhausted initially' is the subsidiary indication.
'month' becomes 'dec' (Dec is an example**).
'up' says the letters should be written in reverse (in down clue: letters go upwards).
'and' means one lot of letters go next to another.
'initially' indicates taking the first letters.
The initial letter of 'exhausted' is 'e'.
'dec' back-to-front is 'ced'.
(Other definitions for cede that I've seen before include "Give up (eg, power)", "Reluctantly give up", "Give up - give way", "Give up (power to another)", "Yield, give way", "Give away", "Yield or surrender", "Give up one's rights", "Give up (territory)" and "Surrender, give up".)