Depression beginning to dog hospital department (4)
I believe the answer is:
dent
'depression' is the definition.
('dent' can be a synonym of 'depression')
'beginning to dog hospital department' is the wordplay.
'beginning to' indicates taking the first letters.
'hospital department' becomes 'ent'.
The initial letter of 'dog' is 'd'.
'd'+'ent'='DENT'
(Other definitions for dent that I've seen before include "Arthur, described by Douglas" , "Hollow from impact" , "Surface hollow" , "Tend to small indentation" , "tooth (13) [FRENCH]" .)