A document referred to the “Pashtoon” people of Afghanistan, which is the spelling used in the applicant’s written submissions. The usual spelling is Pashtun, and quotations from other sources in the document used that spelling.
The advantage of using <u> instead of <oo> is that it’s one less letter. The disadvantage is that the default pronunciation of <u> is /ʌ/, so Pashtun would possibly rhyme with Dunne, whereas the default pronunciation of <oo> is /u:/, so it would definitely rhyme with Doone.
According to Google Ngrams, Pashtoon dates from 1945 and 1953, which is puzzling, given the British wars in Afghanistan from 1839 (maybe they were just Afghans or natives in those days, because there was no reason to distinguish any one group from any other). The two spellings were used about the same until Pashtun became the preferred spelling from the 1980s (the Soviet invasion) and especially 2005 (the US invasion).
Compare Hindoo and Hindu, where there is no ambiguity of pronunciation: <u> at the end of a word can only be /u:/. The two spellings were used about the same until Hindu became the preferred spelling from the 1940s (leading up to Indian independence).
Two more words which spring to mind are igloo (Inuit) and kangaroo (Guugu Yimithirr). Igloo is now linguistically transcribed as iglu, while the first recorded spelling of kangaroo was kanguru (Joseph Banks) and the linguistically reconstructed spelling is gangurru. (Various other spellings were used along the way.) Needless to say, the standard and most common spellings in English are igloo and kangaroo (and Pashtun and Hindu). While the plural of iglu is igluit and the plural of gangurru is gangurru-ngay (Haviland 1979), the plurals in English are igloos and kangaroos (though Linus van Pelt attempted to make igli out of eggshells). Note that in Inuit, iglu refers to any kind of house, while in Guugu Yimithirr, gungurru refers to one specific species of macropod. Also, the people of the Sydney region had no idea what the British were talking about when they used this North Queensland word.
There are also the Chinese names Hu and Hoo, Wu and Woo, which seem to be interchangeable, but for some reason Hoo Jintao looks less presidential than Hu Jintao. Korean 문 can be Mun (ambiguous) or Moon (unambiguous, but possibly causing confusion with the the identically-spelled English word). I once had a colleague with the first name Mun (rhyming with Dunne), who I think was of Malay or Singapore Chinese heritage.