Race in old city run in hospital yard (5)
I believe the answer is:
hurry
'race' is the definition.
('hurry' can be a synonym of 'race')
'old city run in hospital yard' is the wordplay.
'old city' becomes 'ur' (ancient city state).
'run' becomes 'r' (cricket abbreviation).
'in' means one lot of letters goes inside another.
'hospital' becomes 'h'.
'yard' becomes 'y'.
'ur'+'r'='urr'
'h'+'y'='hy'
'urr' going inside 'hy' is 'HURRY'.
'in' acts as a link.
(Other definitions for hurry that I've seen before include "Haste(n)" , "Be quick" , "Rush" , "Look sharp" , "Move with great haste" .)