Prime minister going round each and every London street (4,4)
I believe the answer is:
pall mall
'london street' is the definition.
(I've seen this before)
'prime minister going round each and every' is the wordplay.
'prime minister' becomes 'PM' (abbreviation).
'going round' is an insertion indicator (some letters go around others).
'each' becomes 'all' (similar in meaning).
'and' is a charade indicator (letters next to each other).
'every' becomes 'all' (similar in meaning).
'pm' going around 'all' is 'pallm'.
'pallm'+'all'='PALL-MALL'
(Other definitions for pall mall that I've seen before include "thoroughfare" , "Sixteenth and seventeenth century ball game" , "Monopoly board includes this" , "Old ball game" , "London street with many private clubs" .)