Ran about in frenzy circling Lima's capital (4,5)
I believe the answer is:
ulan bator
'capital' is the definition.
(capital of Mongolia)
'ran about in frenzy circling lima's' is the wordplay.
'in frenzy' indicates anagramming the letters (I've seen this in other clues).
'circling' means one lot of letters goes inside another (circling can mean going around or containing).
'lima' becomes 'L' (phonetic alphabet: alpha, bravo, charlie etc.).
'ran'+'about'='ranabout'
'ranabout' anagrammed gives 'uanbator'.
'uanbator' enclosing 'l' is 'ULAN BATOR'.
(Other definitions for ulan bator that I've seen before include "Mongol city" , "Mongolian city" , "Mongolian capital" , "Capital of Mongolia" , "capital city" .)