I am not a mechanic but I tinker with old cars occasionally. So, when my mother's 2000 Mule 2510 (617cc) began to run poorly and had fuel in the oil they asked me to look at it. I noticed that the choke cable did not work and that the choke was fully on. Since then I have:

Pulled the carburetor and ordered a rebuild kit (amazon). I pulled all the needles and jets and soaked the carcass in cleaner then I blew out the orifices with compressed air. I replaced the float needle, the main jet (was 4 pieces) and the needle jet on the intake side of the carb. The main jet was located in a post next to the float, it is four separate parts and when I reinstalled them I just snugged them down, there did not seem to be any adjustment here. For the needle jet I counted the turns from the bottom and returned the new part to the same place.

Changed the oil, replaced the plugs and fuel filter

Checked for vacuum leaks with carb cleaner.

Started the mule and it would only run with the choke on and only on one cylinder. I measured the exhaust manifold temps and determined that the rear cylinder was not firing, I checked for spark and it was good. So, I pulled the valve cover and the push rods were both out from under the rockers. I checked them and they were not deformed or bent and so I re-installed them and set the valve lash (front too) to .25mm. The rear cylinder fires now but it will only run on full choke and if I attempt to open the choke the mule starts smoking, popping and dies at any throttle position.

I have ordered a fuel pump, it should be here tomorrow. I guess I could replace the carburetor but the only one I could find was on Amazon and of su****ious quality. Any ideas would be appreciated.