City's extremely shady objective overturned by end of May (6)
I believe the answer is:
sydney
'city's' is the definition.
(Sydney is an example)
'extremely shady objective overturned by end of may' is the wordplay.
'extremely' says to hollow out the word (remove centre letters) (only the extremes of the word are used).
'objective' becomes 'end' (both can mean a purpose).
'overturned' says the letters should be written backwards.
'by' is a charade indicator (letters next to each other).
'end of' indicates one should take the final letters.
'shady' with its centre removed is 'sy'.
The final letter of 'may' is 'y'.
'end' back-to-front is 'dne'.
'sy'+'dne'+'y'='SYDNEY'
(Other definitions for sydney that I've seen before include "Last Olympic city" , "Australian city with famous opera house" , "City of Oz" , "- - Carton, 9 hero" , "State capital of New South Wales" .)