He checks out leaders of ill-repute using mayor's public hall (10)
I believe the answer is:
auditorium
'public hall' is the definition.
'auditorium' can be an answer for 'hall' (thesaurus). I am unsure of the 'public' bit.
'he checks out leaders of ill-repute using mayor's' is the wordplay.
'he checks' becomes 'audit' (I can't justify this - if you can you should give a lot more credence to this answer).
'out' is an anagram indicator.
'leaders' suggests taking the first letters.
The initial letters of 'of ill repute using mayor' is 'oirum'.
'oirum' anagrammed gives 'orium'.
'audit'+'orium'='AUDITORIUM'
Can you help me to learn more?
(Other definitions for auditorium that I've seen before include "Performance hall" , "Space for an audience" , "place for concertgoers" , "Space for listeners" , "Body of theatre" .)