Supported by a thesis, the heart of London is improving (2,3,4)
I believe the answer is:
on the mend
'is improving' is the definition.
(on the mend means improving after an illness)
'supported by a thesis the heart of london' is the wordplay.
'supported by' becomes 'on' (something on something else is supported by it).
'a thesis' becomes 'theme' (I can't explain this - if you can you should believe this answer much more).
'the heart of' means to look at the middle letters.
The middle letters of 'london' are 'nd'.
'on'+'theme'+'nd'='ON THE MEND'
Can you help me to learn more?
(Other definitions for on the mend that I've seen before include "Improving in health" , "Making recovery" , "Looking up" , "undergoing recovery?" , "Recovering from an injury or illness" .)