Mope aimlessly around city, acquiring name (6)
I believe the answer is:
eponym
'name' is the definition.
(I know that eponym is a type of name)
'mope aimlessly around city' is the wordplay.
'aimlessly' indicates an anagram (the letters rearranged chaotically or aimlessly).
'around' indicates putting letters inside.
'city' becomes 'ny' (short for New York).
'mope' with letters rearranged gives 'epom'.
'epom' enclosing 'ny' is 'EPONYM'.
'acquiring' acts as a link.
(Other definitions for eponym that I've seen before include "such as Black Beauty" , "a mackintosh, for instance" , "Name derived from title etc." , "handle from place?" , "Derby is a well-known one" .)