In general, Oscar runs shopping precinct in New York (8)
I believe the answer is:
normally
'in general' is the definition.
(I've seen this in another clue)
'oscar runs shopping precinct in new york' is the wordplay.
'oscar' becomes 'O' (phonetic alphabet: alpha, bravo, charlie etc.).
'runs' becomes 'r' (cricket abbreviation).
'shopping precinct' becomes 'mall' (shopping mall).
'in' indicates putting letters inside.
'new york' becomes 'ny'.
'o'+'r'+'mall'='ormall'
'ormall' put within 'ny' is 'NORMALLY'.
(Other definitions for normally that I've seen before include "In the ordinary or typical way" , "Mr Al Lyon? (anag)" , "Ordinarily, usually" , "As a rule, usually" , "As a general rule" .)