Headache ends- in short, Iraq is not even a desert (9)

I believe the answer is:

patagonia

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But I can't explain it at all! Could you help me understand?

(Other definitions for patagonia that I've seen before include "Region of southern Argentina and Chile" , "part of South America" , "Southernmost region of S America" , "far-off land" , "South American region" .)

I've seen this clue in The Hindu.

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