Your cheese is clumping like oil and water.
Okay, it is perhaps oversimplifying slightly, but this is the basic issue--your cheese is made of, essentially, fat and protein (assuming you're using low-moisture mozzarella). The fat and water repel each other, meaning breaking up that ball of cheese is like trying to mix oil throughout a pot of water--it's going to tend to clump together.
What you need is an emulsifier, an agent that will act as a liaison between the cheese and the water to turn it into one smooth mixture. When making sauces, your go-to emulsifier is starch. This is actually quite handy because when you're making pasta you have starch in abundance! When you're done boiling your pasta, just take a bit of the water and add it to your cheese and pasta. With a little mechanical action, you'll see it all turn smooth and even before your eyes.
As far as how much to add--well that depends on how much pasta, how much cheese, how starchy your water is, etc. What I often do is drain my pasta over a bowl, add a ladle-full of water back to my cheesy sauce with the pasta, and stir for a few seconds. It will quickly become clear whether it is mixing well or whether I'll need more water.