I have seen uncited references to Welsh or English longbowmen using falchions, axes, poleaxes, and pikes for close range combat. I'm seeking a more comprehensive understanding of the weapons carried by these medieval bowmen.
The typical archer's sidearm was a sword. Sometimes, they used other weapons too, but the sword appears to have been the most common. Just an ordinary sword - I'm not sure what kind of comprehensive understanding you might be seeking beyond that, so feel free to be more specific.
Our evidence for what weapons they used comes from 3 major sources.
Laws prescribing what weapons must be owned by men in different income/property brackets. E.g., from the Statute of Winchester, 1285:
and from Forty Shillings Land and more, unto One hundred Shillings of Land, a Sword, a Bow and Arrows, and a Knife
Descriptions of battles. E.g., from Chronicles of Enguerrand de Monstrelet, the Battle of Agincourt, 1415:
Their archers, amounting to at least thirteen thousand, let off a shower of arrows with all their might, and as high as possible, so as not to lose their effect: they were, for the most part, without any armour, and in jackets, with their hose loose, and hatchets or swords hanging to their girdles; some indeed were barefooted and without hats.
The English took instant advantage of the disorder in the van division, and, throwing down their bows, fought lustily with swords, hatchets, mallets, and bill-hooks, slaying all before them.
Art. Archers are very often shown wearing swords, and the sword is the most common sidearm archers are shown wearing. Some examples:
https://ugc.futurelearn.com/uploads/images/7e/d9/hero_7ed97256-d757-4a5c-aa73-dcff6f8b265a.jpg
https://i.pinimg.com/736x/30/75/4d/30754d34a0463f279c3e8cac7f778508--th-century-archery.jpg
https://i.pinimg.com/originals/ce/5e/c7/ce5ec77545cca81fac6b1ccaa96b0668.jpg
The sword is an excellent sidearm, since it's easy to wear, and was cheap enough. The English archers who fought in the Hundred Years War (and in many other wars) were respectably-paid professional soldiers, and would have made sure they had an adequate sidearm.