The subject of the inquiry is the bodily function known as the hiccup. While the spelling "hiccough" persists historically and regionally, "hiccup" is the preferred and phonetically accurate spelling in contemporary usage.
However, the most direct link is this: In word puzzle databases, leads to HICCUP. And the alternative spelling hiccough is often clued as “hot” in cryptic crosswords because of a clever three-way link: The subject of the inquiry is the bodily