Burst out: “Clean up half of it” (5)
I believe the answer is:
erupt
'burst out' is the definition.
('erupt' can be a synonym of 'burst out')
'clean up half of it' is the wordplay.
'clean' becomes 'pure' ('pure' can be similar in meaning to 'clean'**).
'up' shows that the letters should be reversed in order (in down clue: letters go upwards).
'half of' suggests halving the letters.
'pure' backwards is 'erup'.
'it' with half the letters taken is 't'.
'erup'+'t'='ERUPT'
(Other definitions for erupt that I've seen before include "be active" , "Burst forth violently" , "Suddenly appear" , "Break out like Etna" , "Break forth" .)