You'll find Midgemong at the top of Blackburn Cove as part of the mission "Symbiosis" that you get from Sir Hammerlock.
Midgemong is a Midget riding an Adult Bullymong and benefits from both for the most part. For the most part, the pair is going to try and ride away from you will attacking with powerful ranged weapons. The pair can take a significant amount of damage (I nearly ran out of ammo) but they are rather predictable once you get the hang of it. They tend to move about the camp and climb up to inaccessible perches to attack you. Stay near the bar that they came out of and when you've figured out where they are going, wait for them to get there -- or shoot at them if they aren't moving along -- shoot at them.
They'll stay there for a good amount of time and be mostly stationary. While they have a ranged weapon, it is largely innaccurate over a medium distance which is the distance from any of the perches to the spot where you start the fight. Wait for them to perch, then shoot. Wait for them to move, perch, and then shoot. This is going to be pretty easy. The tough part are the reinforcements.
Most of the reinforcements are going to spawn below the stairs that this fight starts near. A good portion of these enemies are going to be Marauders and try to shoot you from there but they won't be able to hit you for the most part. Sometimes, you'll get a Psycho or Midget that will come up the stairs after you. These are the only real threat for the most part so just keep your eyes peeled.
Lastly, if you start running out of ammo, There are some vending machines inside including an Ammo Dump. Be careful, however, because when you enter the building, a couple of Badasses will come out of the back door towards you. This can be a very unpleasant surprise when you're very low on ammo so it might be good for you to start the fight out by killing these guys so you don't have to worry about it later.
Just whittle Midgemong down a little at a time as he goes from perch to perch. It might be worth your while to go for the Bullymong first because once the rider is gone, he will move about the area like any Bullymong would except he'll be heavily armored. However, going for the Bullymong may leave you with a heavily armored Midget. The choice is up to you.
