A police department that's sharp (4)
I believe the answer is:
acid
'sharp' is the definition.
(as in sharp-tasting)
'a police department' is the wordplay.
'police department' becomes 'CID' (abbreviation for Criminal Investigation Department).
'a'+'cid'='ACID'
'that's' acts as a link.
(Other definitions for acid that I've seen before include "It turns litmus paper red" , "Etcher's fluid" , "Vinegary" , "So-called drug" , "Sharp - tart - kind of house party" .)