How to separate pups? Great question. You may hear a bunch of responses that will work. My way is a little different. I use a garden hose with a spray nozzle, a large wet dry shop vac, and a large kitchen knife.
Spray the soil away with the hose while sucking up the water and dirt. Keep going till you can see the connection of the pup to the main plant. Once you can see the connection you cut it. You will have a ton of roots exposed. Try to just cut the roots on the pup. If the pup is still stuck in the ground by the roots, cut the roots. It will grow plenty more. If you just horse it out of the ground you have a chance to pull the root out of the corm and invite rot to get started. Once you get done you can just dump the dirt and water back in the hole. Easy.
I call it hydro-vac pup removal.