I've tried preparing my question in a Word document, inserting IPA characters as appropriate, but when I paste the text into the 'Ask Question' box, the text reverts to ordinary script. E.g., a superscript 'h' becomes a plain 'h'.
The easiest way for me to enter IPA is use something like this IPA Keyboard and “type” the right characters there, then use your mouse to copy it from that bottom window to the entry widget on ELU. It’s nice because it handles all IPA, including arbitrary combining diacritics.
It sounds like you were using Word's superscript formatting. There is actually a special unicode character for IPA superscript h: the "ʰ" character, so you may want to use that instead. This character should always appear superscript no matter how it is formatted.
You can find characters like this in the Windows Character Map if you are using Windows; most operating systems have an equivalent utility.