Incompetent appeal collecting very little money in NY without permission (7)
I believe the answer is:
illicit
'without permission' is the definition.
'incompetent appeal collecting very little money in ny' is the wordplay.
'incompetent' becomes 'ill' (as in ill-advised or ill-informed).
'appeal' becomes 'it'.
'collecting' means one lot of letters goes inside another.
'very little money in ny' becomes 'ic' (resembles 1c or one cent - money in the US).
'ill'+'it'='illit'
'illit' placed around 'ic' is 'ILLICIT'.
(Other definitions for illicit that I've seen before include "Not lawful" , "Not approved by law" , "wrong" , "Without approval" , "completely banned" .)