Desert irritating director having previously noted big mouth (4)
I believe the answer is:
gobi
'desert' is the definition.
(I know that Gobi is a desert)
'irritating director having previously noted big mouth' is the wordplay.
I cannot quite see how this works, but
'mouth' could be 'gob' (informal term for the mouth) and 'gob' is found within the answer.
A single letter 'i' remains which might be clued in a way I don't understand.
This may be the basis of the clue (or it may be nonsense).
Can you help me to learn more?
(Other definitions for gobi that I've seen before include "Mongolian/Chinese desert" , "Mongolian expanse" , "Desert in central Asia" , "The . . . . Desert is a barren plateau of S Mongolia and N China" , "Desert, mostly in Mongolia" .)