Do the melungeons ever claim heritage from the Welsh?
I think the melungeons claim to be Portuguese.
The primary researcher discovered almost immediately that the Spanish ran a POW camp in South Carolina. They took prisoners there who they'd captured in battles in the Balkans or the Eastern Mediterranean.
In those times it was typical for the Spanish to simply dispose of any Moslems they captured as being ineducable. On the other hand, although the Roman and Greek churches were at odds in the 1500s, any Orthodox soldiers or sailors taken prisoner probably made up the greater part of those folks who went to the POW camp.
However, there's evidence that the Spanish didn't just kill all the Moslems because virtually every Melungeon checked so far has a "marker DNA sequence" unique to themselves and a group of descendants of black slaves brought to Karachi in what is now Pakistan way back in the 1200s or 1300s. Somehow that group's genetic marker got to the Americas. It's hypothesized that a group of these folks in Karachi were taken to the Middle East by the Turks and they ended up in the Turkish Navy or maybe an Army unit somewhere in the Balkans where they could encounter the Spanish or other Christians in any one of a number of wars that went on in that period.
The focus is on the Melungeons having their origins in America as an identifiable special population sometime in the 1500s. The Melungeons also have considerable European ancestry, and American Indian forebears. Both may well derive from intermarriage with Eastern Cherokee, or Iroquois, or maybe even Ojibway-Chippewa themselves (with a whole huge Sa'ami component).
Always remember, the Cherokee were a mixed people themselves, and there were those who were very light and those who are very dark. They were just like today's Americans ~ some light, some dark, some inbetween and frequently with no idea how they got that way.
Prince Madoc derives from centuries before the time of the Melungeon formation.
BTW, because people met European looking "Indians" from the earliest times there was always this ambition to explain it with "recent" European contact ~ not Ice Age contact. Now that we know there was Ice Age contact, the questions have changed.
Not that I am aware. “Portugee,” Turk and/or English Lost Colony descendants in addition to native tribes is the usual claim, among the Melungeons and several mixed race tribes such as the Lumbee. Some are more forthcoming about a degree of African ancestry than in the past as well, but not all are. Technically speaking, they’re regarded as being “tri-racial isolate groups.”