In a week or so city goes to ruin (6)
I believe the answer is:
decays
'goes to ruin' is the definition.
(to decay is to become ruined)
'in a week or so city' is the wordplay.
'in' indicates putting letters inside.
'a week or so' becomes 'days' (a week is 7 days).
'city' becomes 'EC' (the City of London postcode).
'days' placed around 'ec' is 'DECAYS'.
(Other definitions for decays that I've seen before include "Rots, moulders away" , "Rots, disintegrates" , "Decomposes or falls into ruin" , "Rots and decomposes" , "Goes off" .)