Threw the lid off the organ (4)
I believe the answer is:
lung
'organ' is the definition.
(I know that Lung is a type of organ)
'threw the lid off' is the wordplay.
'threw' becomes 'slung' (I've seen this in another clue).
'the lid off' means to remove the first letter (in a down clue, lid can mean the top letter).
'slung' with its first letter taken off is 'LUNG'.
'the' acts as a link.
(Other definitions for lung that I've seen before include "Breathing apparatus" , "Respiratory organ" , "One of a pair of organs in the ribcage" , "breather" , "Organ used for breathing" .)