The strongest army before the hand held gun

So,wich one?

I think it more of a halberd, than a pike.

I would go with the passage of English 100 Years War Army or Catalan Company, you know that knights with Almuagavears (I know i spelled that wrong) were devastating.

The French knights at Agincourt and Crecy were moving targets, it didn't do them much good. 10,000 longbowmen can launch 100,000 arrows per minute. It would be like being in a heavy shower and hoping every single raindrop misses you.

At the battle of Agincourt 7,000 English beat 40,000 French. 10,000+ French dead, 30+ English dead. At the Battle of Towton there were 60,000 English present. Plenty to take out a couple of hundred thousand Mongols?

Supply indeed would be a problem. A good reason for an English army not to go to the middle east or the steppes but stick to Europe where it can fortify it's supply line with wooden forts and by taking castles and walled towns. And if things got desperate live of the land from farms, villages and towns. As Napoleon did.

Then again Richard I took an army to Jerusalem and kept then supplied. And Alexander found himself with no supply lines whatsoever and made it to Alexandria living off the land, simply by careful planning of his route via major towns nd cities.

The Mongols themselves would suffer from terrible supply problems in Europe. The towns being fortified, so they would need to seige them to get their supplies. They couldn't bypass castles either, or they'd lose their own supply line so would find themselves having to siege every castle they came across. Something they would find very difficult.

The real difference between the Asiatic composite bow and th longbow is not so much range but deteriation of accuracy. The Asiatic is actually much more advanced than the longbow which has existed viirtually uncanged since pre-historic times. But the Asiatic is really only a makeshift solution of the problem lack of good trees.

The longbow as we know stays true at long ranges, a unit of men at 200m is easily within it's capacity. This is the reason it's an infantry weapon, to fire accurately at that sort of range a steady platform is required. The Asiatic long was only accurate up to about 60 metres, after 60 metres the accuracy curve starts to drop off. Horses are not steady platforms, so any sort of long range shooting not possible anyway.

English armies of the day tended either to deploy on top of hills, giving them a 360 degree round view. On poor quality or boggy terrain that's extremly difficult to move across, disordering infantry and making cavalry ineffective or positions where they could anchor themselves to a piece of impenetrable terrain the opponent couldn't get around.

For an opponent cavalry to ride around behind it take minutes, for a longbow unit to about face takes seconds.