Motorist runs over a dog outside hospital (4,3)
I believe the answer is:
road hog
'motorist' is the definition.
(I have seen 'Inconsiderate motorist ' mean 'road hog' so perhaps 'motorist' could also mean 'road hog')
'runs over a dog outside hospital' is the wordplay.
'runs over' becomes 'ro' (this could be a standard abbreviation I've not previously seen).
'outside' means one lot of letters goes inside another.
'hospital' becomes 'h'.
'dog' placed around 'h' is 'dhog'.
'ro'+'a'+'dhog'='ROAD-HOG'
(Other definitions for road hog that I've seen before include "Reckless motorist" , "Discourteous driver" , "Mr Toad, for one" , "Thoughtless driver" , "Selfish and dangerous driver" .)