YorkshireYankee wrote: Child Ballad 31, "The Marriage of Sir Gawain", in which King Arthur is required to find out "what women most desire": Text of the original ballad
It has hundreds of verses (though not all have survived) and unfortunately, there is no known tune.

It's worth noting that "The Marriage of Sir Gawain" is from the Percy Folio, and it is never safe to assume that an item in the Percy Folio is a traditional song. Some of the contents are traditional songs -- but a great many items in the Folio are metrical romances that have been chopped down. I already cited (October 5) the instance of "The Squire of Low Degree," where the Percy Folio version has chopped out about five-sixths of the original romance.

In the case of "Sir Gawain," we even have the original romance, or something very like it: "The Marriage of Sir Gawain and Dame Ragnall." The generic "Loathly Lady" tale is even older, since we have two other English versions (Gower's "Tale of Florent" in the Confession Amantis, which is far inferior to "Ragnall," and Chaucer's "Wife of Bath's Tale," which perhaps slightly superior in plot and of course vastly superior in language).

There is a whole book about the Loathly Lady, S. Elizabeth Passmore, The English Loathly Lady Tales: Boundaries, Traditions, Motifs (Studies in Medieval Culture XLVIII), which I have to get around to reading one of these days. On "Sir Gawain" and the relationship between the English Loathly Lady tales, you can find a lot more on the Ballad Index entry on "Sir Gawain," http://balladindex.org/Ballads/C031.html.