The besetting sin of the Greens? (4)
I believe the answer is:
envy
'sin of the greens?' is the definition.
'envy' can be an answer for 'sin' (envy is a kind of sin). I am not certain of the 'of the greens?' bit.
I don't understand the rest of the clue.
Can you help me to learn more?
(Other definitions for envy that I've seen before include "Covet; be jealous of" , "Desire of another's property etc" , "One of seven deadly sins" , "The green-eyed emotion" , "Jealous longing" .)