Well, the effective range and the maximum range are two different questions, as are, "effective range while moving at speed" and "effective range while stopped," and "Effective range while dismounted." Beyond that, the horse archer was effective beyond the range of his bow and arrow. That is the question I will attempt to answer

There are not many definitive answers to the maximum or even effective range of a Mongolian horse archer. Part of this, I believe, is due to how the Mongolians thought about horse archer tactics.

The concept of "effective range" as a definite number is more of a standard military idea than a Mongolian military idea. Range matters most when your ranged units are stationary and have to remain stationary to shoot. To the English longbowman, range matters a fair amount- before he can shoot, he has to get to a place where he can shoot, draw a very large bow, aim, and release, either with a lot of other archers while firing in volleys, or while shooting at specific targets. Typically, he is not fighting on his own, as he would be behind a line of spears or behind fortifications. The bulk of the conflict would be hand-to-hand at close range, where armored men with spears, axes, swords, and clubs would brawl.

So, his effective range is the farthest away he can be while still being useful while the armored guys fight it out. A good definition of effective range for us, then, is, "The range at which a particular unit is useful."

But Mongolians did not fight that way. The bulk of the Mongolian military consisted of mounted horse archers, supported by some mounted lancers. Sometimes, they would use Chinese-developed siege weapons. So, when asking about the effective range of a horse archer, we need to think about the range of the entire horse/bow system, because that determines the range we are trying to determine.

The horse archer was the "front line" warrior in the Mongolian military. Mongolians won battles mostly through superior maneuverability and by picking the "correct" battles to fight. Many times, they would push forward, feign an attack, retreat, and then double back, circle the enemy, and flank them. Typically when making these types of movements, the horse archers would be a few hundred feet away when they released arrows. Sometimes they were aimed, and sometimes they were in volleys. All of the horse archers were excellent shots who could pick targets, but sometimes tactical situations would call for unaimed shots.

For example, if the warriors were trying to lead a target into an ambush, they might ride forward, shoot into a line of infantry, turn, and ride away, knowing that skirmishers or cavalry would pursue. The arrows and bows were not effective against focused infantry lines, especially lines of infantry armed with spears. Instead of trying to whittle away defended positions, the Mongolians would find ways to scatter formations and cause confusion among the enemy ranks. So, after the skirmishers or light cavalry started to follow the Mongols in pursuit, the Mongols would ambush, destroy the pursuers, return to the formation they were attacking, and then do the same thing. While the formation of infantry or whatever else attempted to deal with one group of warriors, another group would make a wide flanking movement and start attacking from behind- they would attack with bows, but it was entirely possible that they might then close distance and attack with lances and swords before falling back and attacking with bows again.

And this could be happening across miles of pasture and space. A horse archer could move between 50-70 miles in a day.

So, for a strategic range, I would say, probably 50 miles. For a tactical range, I'd estimate about 1000 meters. Note- that is not the firing range, which was likely 300 meters, depending on the quality of the bow and the archer.

Take a gander at this thread.