City, within a week or so, deteriorates (6)
I believe the answer is:
decays
'deteriorates' is the definition.
(I've seen this in another clue)
'city within a week or so' is the wordplay.
'city' becomes 'EC' (the City of London postcode).
'within' means one lot of letters goes inside another.
'a week or so' becomes 'days' (a week is 7 days).
'ec' going inside 'days' is 'DECAYS'.
(Other definitions for decays that I've seen before include "Rots and decomposes" , "Goes off" , "Decomposes or falls into ruin" , "Rots, disintegrates" , "Rots, moulders away" .)