I believe people are right about the salts causing the scumming, but I think they're not quite right about the causes. I have found this scumming very consistently in areas that are touched heavily while in a firmer, less-plastic state. The best prevention is trying not to touch the sculpture once it is progressing into leather-hard and beyond. I'm not sure if possibly heavily re-wetting where you've touched makes any difference.
My belief is that as the clay begins to dry it is water-hungry -- in fact, it appears more hydrophilic at that stage then when it is bone-dry. When you then press your skin into the sculpture for a prolonged period the clay happily sucks oils(which must contain some salts) from your hands. When fired these salts scum to the surface. If you're lucky, there can be so many of them that they can be flaked off. If it's a small amount just hazing an area you can't really remove without sanding and destroying your work in part.
For example, I often get scumming on the tips of noses when hollowing heads. I cut the head in half and when I'm hollowing out the front half the nose is the driest part and is often under the most intense contact with my hand as I hold it. The only solution is to try to put some kind of barrier between your hand and the sculpture at such stages.
Fortunately I have found most finishes remove these spots, particularly if they are the slight, hazier sort. Wax finishes have always removed them for me in that case.
I have no experience with glazes but I assume from the above reports that a glaze will often fix the problem if it's not severe enough to be a solid mass of scum (which can be flaked off, usually, besides).
An easy test, for me, of whether the scum will show-through a finish or not is to add water to the area. If the scum disappears when wet, then it will likely disappear when waxed (disappears when oiled too in my experience but the oil usually dries out eventually and the spot comes back).
I *DO NOT* believe the problems are from your water or your clay body. I have had this problem with a wide variety of water and 4 diferent clay bodies. The scumming ALWAYS showed up precisely where I touched the clay, not randomly.
I believe hte key in avoiding these spots is to not work dryer portions by hand and to be very cautious when lifting and moving sculptures in the drying process (perhaps wearing gloves or perhaps washing hands first). There is a very particular stage at which the clay is prone to doing this as a result of hand contact and once you get used to that you can learn to handle it lightly if at all during that stage.
I doubt any amount of spraying-down after touching would help much as the oils have probably already been sucked into the body. Prevention is probably the only option. Fortunately, in my experience, you can see where the white spots will be in the bone-dry stage if you look very closely. If yours are not showing up when bone-dry you may be experiencing something different.