Toilet leaking out of handle hole; pump spits water out of top
The seal at the top of the valve has failed. Normally a small portion of the incoming water should go through that small tube off the side, while the rest of it should come out the bottom of the valve. None of it should be sprayed about like that.

If it were my toilet I'd see if I could take that top off and try to fix the seal, but I wouldn't try very hard, because it looks like a cheesy valve, and new ones are around $10-$15. If it were my landlord's toilet I'd tell the landlord about it. A good way to prompt some action would be to note that the toilet is leaking and probably causing damage to the floor.