I assume marshmellow is just people who've never heard of a marsh mallow plant and have misheard the pronunciation, then more people hear it, and it persists.
Haitch/aitch is actually a bit of a minefield.
I was brought up to say aitch, and taught haitch was flat out wrong. Very firmly corrected if I said haitch.
I've since found out that, particularly in Ireland, it was historically a marker of Protestant / Catholic, and used as a 'tell' when working which side of the divide someone was on. It seems to have persisted across England too, presumably because the monarch, (being Protestant) would have said aitch, and haitch was then regarded as 'wrong' in comparison. It's really very interesting, worth a Google.
I've found myself correcting DD to aitch, when she had a teacher that said haitch. It's genuinely like fingernails on a blackboard to me, even worse than f instead of th. I guess that conditioning from an early age for you!
I'm culturally CofE, going back many generations, I guess it's something so intertwined with that, that it's instinctive to correct it to what parents, grand parents, and so on back, taught.