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' (abbreviation for Ear, Nose and Throat department).
'opened by' means one lot of letters go next to another (some letters open or go before others).
'king' becomes 'k' (abbreviation for king in chess).
'ent' 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" .)