Green stuff eaten (not “ate”) up in desert (5)
I believe the answer is:
negev
'desert' is the definition.
(I know that Negev is a desert)
'green stuff eaten not ate up' is the wordplay.
'green stuff' becomes 'veg' (I can't justify this - if you can you should believe this answer much more).
'not' suggests deleting specific letters.
'up' shows that the letters should be reversed in order (in down clue: letters go upwards).
'eaten' with 'ate' taken away is 'en'.
'veg'+'en'='vegen'
'vegen' reversed gives 'NEGEV'.
'in' is the link.
Can you help me to learn more?
(Other definitions for negev that I've seen before include "part of Israel" , "Wilderness" , "Israeli desert area abutting Sinai" , "Southern Israeli desert" , "Desert region in Israel" .)