Anoushka is Russian, a pet form of Anna, pronounced ah-NOOSH-kah. In Hungarian it's Annuska, in Czech Anuska, both pronounced the same. There's also the Czech Anezka, pronounced ah-NEJH-kah, a form of Agnes. In Eastern European languages I've never heard of Ah-na-ush-ka but it seems to me like what someone unfamiliar with the name might make of Anoushka, 'sounding out' all the vowels instead of using the diphthong for ah-nuh-ush-ka. A clue as to where you heard it might lead in one direction or the other.
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