Criminal’s bad luck, dropping half the rings (4)
I believe the answer is:
hood
'criminal's' is the definition.
(hood is a kind of criminal)
'bad luck dropping half the rings' is the wordplay.
'bad luck' becomes 'hod' (I can't justify this - if you can you should believe this answer much more).
'dropping' is an insertion indicator.
'half the' indicates taking half.
'rings' becomes 'oo' (multiple circles).
'oo' cut in half is 'o'.
'hod' placed around 'o' is 'HOOD'.
(Other definitions for hood that I've seen before include "Cover < the upper part < of tough" , "this could provide cover" , "Gangster - cover for head of falcon" , "academic garment" , "Cowl -- violent criminal (slang)" .)