It just so happens I have "100 English Folksongs, edited by Cecil Sharp", sitting by me as I noticed your thread. If you don't know, Sharp is one of the leading figures in the world of folk music. It is no coincidence that the English Folk Dance and Song Society is based in a building called Cecil Sharp House. Here is his notes for the song Henry Martin, which might be of some use to you.
"Versions of this ballad with tunes, are in Kidson's "Traditional Tunes" and "Songs of the West". The words appear on a Catnach broadside; and in Percy's "Relkiques".
In "English and Scottish Ballads", Chil prints the versions in Traditional Tunes and Songs of the West, and gives, in addition, four other sets - one from Motherwell's MS., two traditional copies from residents in the U.S., and a Suffolk fragment contributed by Edward Fitzgerald to "Suffolk Notes and Queries (Ipswich Journal, 1877-78)".
In several versions, the hero is variously styled Henry Martin, Robin Hood, Sir Andrew Barton, Andrew Bodee, Andrew Bartin, Henry Burgin and Robedrtson.
Child suggests that "the ballad must have sprung from the ashes of'Sir Andrew Barton' (Percy's Reliques), of which name 'Henry Martin' would be no extraordinary corruption." The Rev. S. Baring-Gould, in his note to the ballad Songs of the West, differs from this view and contends that the Percy version is the ballad "as recomposed in the reign of James I, when there was a perfect rage for re-writing the old historical ballads".
I am inclined to agree that the two versions are quite distinct. Sir Andrew Barton deals with the final encounter between Barton and the King's ships, in which Andrew Barton's ship is sunk and he himself killed; whereas the traditional versions are concerend with a piratical raid made by Henry Martin upon an English merchantman. It is true that in Songs of the West, Henry Martin receives his death wound, but, as Child points out, this incident does not square with the rest of the story and may, therefore, be an interpolation.
Unlike so many so-called historical ballads, this one is really based on fact. In the latter part of the 15th century, a Scottish sea-officer, Andrew Barton, suffered by sea at the hands of the POrtuguese, and obtained letteres of marque for his two sons to make reprisals upon the trading ships of Portugal. The brothers, under the pretence of searching for Portuguese shipping, levied toll upon English merchant vessels. King Henry VIII accordingly commissioned the Earl of Surrey to rid the seas of the pirates and put an end to their illegal depredations. The earl fitted out two vessels and gave the command of them to his two sons, Sir Thomas and Sir Edward Howard. They sought out Barton's ships, the Lion and the Union, fought them, captured them, and carried them in triumph up the river Thames on August 2, 1511.
I have noted down in different parts of England no less than seventeen variations of this ballad."
I think you can safely say from that Adam that Child did not write this song (if indeed Child actually wrote any song he collected, for that matter).