Hi Gensnr!
My notes on the Pitfirrane (etc) estates and their workers only start at 1696, so I don't know anything about people before that date. There aren't any Kinghornes listed, but quite a few Andersons - maybe related, maybe not - it's a fairly common name.
Among the miners I have Charles Anderson in 1696, 1706/10/12, Charles (younger) in 1712, William and Adam in 1724, Charles and David in 1729, David, Charles, William and Adam in 1731.
In 1732 Adam, William, Charles, David and Robert are listed, but Adam Anderson died in Oct 1732.
William "& his wife & bairn" ran away, but were recaptured and brought back in Jan 1733. William again ran away, but was brought back "when trying to go to H.M. Plantations" in 1733.
The last mention of any coalminer Andersons is John, in 1734, but Adam is shown as an "oversman" in 1743, and William and Robert Anderson are noted as blacksmiths in Crossford village in 1742: maybe they were the same as the earlier miners with these names, maybe not.
As to whether the Burts belonged to a "clan" - I don't think they'd have recognised such a thing, living as they did in lowland Scotland, not the Highlands. Maybe the Burt family has been awarded an alleged "clan tartan" but, if so, that will simply be the product of someone's imagination, going back, like so much of this clan tartan nonsense, to the mid-19th century - at the very earliest.
Harry.