Terribly chilly at Mayor of London's place (4,4)
I believe the answer is:
city hall
'mayor of london's place' is the definition.
Although both the answer and definition are singular nouns, I can't see how they can define each other.
'terribly chilly at' is the wordplay.
'terribly' indicates an anagram.
'chilly'+'at'='chillyat'
'chillyat' is an anagram of 'CITY HALL'.
Can you help me to learn more?
(Other definitions for city hall that I've seen before include "local government building" , "Local authority" , "In the US, municipal offices collectively" , "US municipal government building" .)