There is organic material (ie paper pulp) in the paperclay, and mold tends to form quite readily even in unopened bags of paperclay. I have used regular laundry bleach, Pine Sol, Mr. Green, etc as disinfectants to retard the growth of the organisms. I'm partial to Pine Sol as I find it's less harsh on my hands and I like the scent of the pine.
Normally one capful of the disinfectant in your bucket of paperclay slip will do it. It will retard the growth of the mold for a while, but eventually, the mold will come back. The thing to do is mix as much paperclay slip as you need at one time. However, I find this is difficult to judge as I also like to have extra on stand by. I also add a few drops (maybe 1/2 capful) into my spray bottle.
I get my paperclay commercially so it really depends on how long that batch has been sitting in the warehouse before it becomes mine. Sometimes I get really fresh paperclay - nice and clean, other times, there is already black mold starting inside the bag. It is surface mold, the inside is still good. The property of the paperclay is not affected by the mold, other than the "icky/yucky" factor.
From both the drying effects of the clay and the disinfectants I use; not to mention the frequency of thorough hand washing with antibacterial soap and water, my hands take a beating. Frequent application of a hand lotion of your choice will help hydrate your hands and make them feel better, even in between breaks from working with your clay. Some people wear gloves. That is also an option. I have to admit I've never seen anyone throwing on the wheel with gloves on! Working with sanded/hi grogged clay will "eat up" those flimsy things in no time. Moreover the "fun" factor of working with clay is the ability of feel this wonderful, yet very simple, almost primordial material.
Ahh .... the price we pay for our passion.
DISCLAIMER: Keep food and drinks away from your work area. Always throughly wash/scrub your hands and other areas of your body covered with clay after your work and before handling/preparing food. This is just plain common sense.