Crossword Genius

Manage US city with medical service, ultimately in place of political agreement (9)

Ross

I believe the answer is:

runnymede

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'place of political agreement' is the definition.
(where the Magna Carta was signed)

'manage us city with medical service' is the wordplay.
'manage' becomes 'run' (both can mean to organise).
'us city' becomes 'NY' (short for New York).
'with medical service' becomes 'mede' (I can't explain this - if you can you should believe this answer much more).
'run'+'ny'+'mede'='RUNNYMEDE'

'ultimately in' is the link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.

Can you help me to learn more?

(Other definitions for runnymede that I've seen before include "Scene of Magna Carta's signing" , "place on the Thames" , "where barons got the concession" , "Magna Carta meadow" , "Place where Magna Carta was signed" .)

I've seen this clue in The Times.
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