Day lodging — part of London in decline (5)
I believe the answer is:
decay
'london in decline' is the definition.
'decay' can be an answer for 'decline' (synonyms). I'm unsure of the 'london in' bit.
'day lodging part' is the wordplay.
'lodging' becomes 'ec' (I can't explain this - if you can you should give a lot more credence to this answer).
'part' indicates putting letters inside (the inserted letters split open or part the word).
'day' enclosing 'ec' is 'DECAY'.
'of' is the link.
Can you help me to learn more?
(Other definitions for decay that I've seen before include "Briefly fashionable" , "Decline, decompose" , "Deterioration" , "Decompose or fall into ruin" , "Putrefy" .)