But the question was how to pronounce “Hagia Sophia” in English (presumably in American English). Obviously in Greek or Turkish or Klingon it would be pronounced differently – but the question as “in English.”
The use of pronunciation forms from the original language is (IMHO) pretentious and unjustified. There is an assumption that if the word is said in the same way as the original context that is some how more authentic. And it just isn’t so. When we (Americans) speak of that large former church in Turkey, we have a set of concepts that are distinctly 20th century, Western European, and American in flavor. The Greek, the Turk and the New York might all mean the same physical place – but our associations and deep meanings will be drastically different. It is sophistry to use the “authentic” pronunciation, as if that made the meaning more correct.
In English it probably should be “Holy Wisdom” church/museum. If speaking of the international landmark, it becomes a proper name used in English (so Schwartz is not changed to Black) and the place should be “Hah-gi-ah So-fee-ah.” The Greek pronunciations should be saved for a direct quotation in Greek (and then when translated into English – the version with the “H” sound should be used).
My grandfather came from Deutschland – but my family roots are in Germany.