County hospital department opened by king (4)
I believe the answer is:
kent
'county' is the definition.
(Kent is an example)
'hospital department opened by king' is the wordplay.
'hospital department' becomes 'ENT' (Ear, Nose and Throat department).
'opened by' is a charade indicator (letters next to each other) (some letters open or go before others).
'king' becomes 'k' (chess abbreviation).
'ent' put after 'k' is 'KENT'.
(Other definitions for kent that I've seen before include "Garden of England - Superman's family name" , "Earl Lear banishes" , "known to Scots" , "English architect and landscape gardener" , "County, garden of England" .)