Good question! There's no direct evidence to indicate that English mounted archers shot from horseback. Now, medieval chroniclers were as a rule not interested in recording minor tactical details, so there's no evidence to say that English mounted archers never shot from horseback either, but if it happened, it would have been a rare occurrence. Mounted archers sometimes fought from horseback, but seem to have used their swords to do so.

There's two significant aspects of mounted archers that distinguished them from foot archers. One, the fact that they could afford a horse meant that they were from a higher social and financial class than foot archers. This in turn meant that they could afford better equipment (sword and buckler, perhaps some pieces of armor, etc.). The presence of these men among the ranks of the bowmen allowed the English a great deal of tactical flexibility. When their archers ran out of ammunition (or the enemy had gotten too close to shoot), the archers could drop their bows and spread out to flank the enemy or reinforce the men-at-arms in critical sectors. They could also perform tasks like guarding prisoners or finishing off enemy wounded, thus freeing up men-at-arms for the main fight. Mounted archers were also capable of quickly running back to their horses and jumping on their mounts to participate in a charge, like in the Battle of Poitiers.

Two, mounted archers were of great importance on campaign. Although they could not outfight any significant enemy presence, mounted archers could perform the traditional tasks of light cavalry: capturing prisoners for interrogation, scouting, raiding villages for food and supplies, etc. In a world without modern communications gear, a dude on a horse was the best possible method for getting information from one place to another. Many battles of the HYW were preceded by several days of cavalry patrols running into each other as both sides attempted to figure out where the enemy was and what he was doing. Having a large number of mounted archers relieved the mounted men-at-arms of performing all of these tasks on their own.

Many popular takes on the HYW emphasize the gap in rank between aristocratic French knights and the common English bowmen they fought. However, the gap between men-at-arms and archers was nowhere near as large a gulf as is often assumed. The mounted archers were sometimes the second sons of families who couldn't afford to equip more than one son as a man-at-arms. Some men might show up to one campaign as an archer and appear as a man-at-arms for the next, if he was particularly successful at looting. Although the mounted archers would fight on the battlefield in a similar manner to the foot archers, they were recorded as a separate group because of both tactical and social distinctions.