Aspires to be terribly posh around Mile End (5)
I believe the answer is:
hopes
'aspires' is the definition.
(I've seen this in another clue)
'terribly posh around mile end' is the wordplay.
'terribly' indicates an anagram.
'around' is an insertion indicator.
'end' indicates one should take the final letters.
The final letter of 'mile' is 'e'.
'posh' is an anagram of 'hops'.
'hops' going around 'e' is 'HOPES'.
'to be' is the link.
(Other definitions for hopes that I've seen before include "Believes that good things may happen" , "Expectations" , "Desires and expects" , "Aspirations, wishes" , "Aspires" .)