Police officers fill their bellies and throw up (8)
I believe the answer is:
disgorge
'throw up' is the definition.
(I know that throw up can be written as disgorge)
'police officers fill' is the wordplay.
'police' becomes 'dis' (detective inspectors or DIs).
'officers fill' becomes 'gorge' (I am not sure about this - if you are sure you should believe this answer much more).
'dis'+'gorge'='DISGORGE'
'their bellies and' acts as a link.
This may not be right. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for disgorge that I've seen before include "Throw up, regurgitate" , "So rid egg? (anag)" , "yield under pressure" .)