If we're talking about English longbowmen in the high Middle Ages, then yes, they would have had some sort of melee weapon for backup use if they were surprised at close ranges. Jean Warrin, writing about Agincourt, described the actions of the English longbowmen as they ran out of arrows and charged the French flanks: “the English archers...then took their swords, hatchets, mallets, falcon-beaks and other weapons...”

The mallet in particular seemed to be a common tool of the archer, since the English preferred to fortify their positions with sharpened stakes to protect themselves from cavalry and these mallets were useful to pound stakes into the ground. As Religieux of St. Denis (Michel Pintoin) said, "In addition, many of them [archers] had adopted a type of weapon until then unknown - great lead-covered mallets [clavam plumbeam] from which one single blow on the head could kill a man or knock him senseless to the ground."

It would be worth noting that while this worked out pretty well for them at Agincourt, this backup weaponry in no way made an archer the equal of an infantryman. In the battles of Patay and Verneuil, the English longbowmen were routed by heavy cavalry before they had a chance to set up their protective stakes.

I recommend you check out this post by /u/MI13: https://www.reddit.com/r/AskHistorians/comments/4uhwmf/meta_what_kind_of_questions_do_you_guys_like_to/d5q5k9p/

It's not really correct to think of medieval soldiers as being "issued" or "receiving" weapons and training in the way that members of modern militaries do. Medieval armies were largely ad hoc organizations formed from disparate parts, not permanent bureaucracies that induct, train, equip, and deploy men in cohesive units with standardized weaponry. For most soldiers, this meant that you were supplying your own equipment and training; for members of retinues, it meant you were getting what your lord thought you should have.

Regardless, we have ample evidence that English archers of the Hundred Years War were quite well armed and fully capable of fighting as light infantry when called upon to do so. Wealthier archers would have worn mail, brigandines or bits of plate armor; poorer archers would at least have helmets and padded jacks. Swords and bucklers were quite common, along with axes, hammers, and various pole arms. Now, would archers stand up to heavy infantry and/or heavy cavalry unsupported? No; and when the English heavy infantry wasn't present to take the brunt of the head-on fighting, they fared very poorly indeed.