The story of one doing deeds requested of him by an animal, ghost, etc., and being rewarded by a transformation of the apparition, is an old one, and "King Henry" is an excellent telling of the tale.

Bronson said " Mrs. Brown of Falkland is again the sole preserver of this offshoot of Arthurian romance. Her tune has been saved in two manuscripts, the "abbotsford" manuscript of Scottish songs in Scott's library, and William Tyler's manuscript at Aldourie Castle ....."

In other words, variations and mutations would be those of more modern singers such as those listed by Susan of DT.

A stray thought- would the story of King Henry II and the Miller of Mansfield perhaps provide inspiration for a variant, possibly involving cuckoldry?
This ballad is in the Bodleian Library, Harding B 2(82). I couldn't find it in the DT.