Get better after leaving city with traveller (5)
I believe the answer is:
rover
'traveller' is the definition.
(rover is a kind of traveller)
'get better after leaving city' is the wordplay.
'get better' becomes 'recover' ('recover' can be a synonym of 'get better').
'after leaving' suggests deleting specific letters.
'city' becomes 'ec' (the City of London postcode).
'recover' with 'ec' taken out is 'ROVER'.
'with' acts as a link.
(Other definitions for rover that I've seen before include "rambler" , "Pirate; dog's name" , "Person given to wandering" , "Car - dog" , "Wanderer, rolling stone" .)