Half believed police department's tapped judge (6)
I believe the answer is:
decide
'judge' is the definition.
(decide can mean to make a judgment)
'half believed police department's tapped' is the wordplay.
'half' indicates taking half.
'believed' becomes 'deemed' (eg you believe or deem something to be worthless).
'police department' becomes 'CID' (abbreviation for Criminal Investigation Department).
'tapped' indicates putting letters inside (tap can mean to take or import).
'deemed' cut in half is 'dee'.
'dee' placed around 'cid' is 'DECIDE'.
(Other definitions for decide that I've seen before include "Adjudicate" , "Reach a judgment" , "Judge" , "Resolve or give a judgment" , "Conclude" .)