Argue about your old mayor? (6)
I believe the answer is:
worthy
'mayor?' is the definition.
(I've seen this before)
'argue about your old' is the wordplay.
'argue' becomes 'row'.
'about' shows that the letters should be reversed in order.
'your old' becomes 'thy'.
'row' back-to-front is 'wor'.
'wor'+'thy'='WORTHY'
(Other definitions for worthy that I've seen before include "big cheese?" , "Good enough" , "One of the distinguished people" , "Somebody" , "Meriting respect or reward" .)