Care and cleaning of your bogu varies depending on what the actual damage is. Given the surface of the do is generally bamboo with lacquer over it, you essentially wind up with three sorts of stains. First, and the usual item with white marks, is salt from sweat. But that usually gets cleaned off readily with a wet towel, so the odds are good that that's not your problem. Secondly, there's darker stains from the shinai oil. That's usually cleanable afterwards with the wet cloth, but can sink in if you don't regularly clean your equipment afterwards. I don't have a good solution for that one, but it's darker stains, so that's probably not your problem. Lastly, and most likely, there is scuffs and residue from strikes. The do, as you know, is covered by layers of lacquer, typically opaque colors covered with clear lacquer. Your first step is a scratch test. A fingernail will work fine if you have longer ones. A scrub brush is a reasonable alternative. Can you scratch the mark off in the same way you'd clean off a shoe-sole scuff on a hardwood floor? If so, you're looking at more scrubbing and picking to get the material off. For a white mark, you're more likely looking at the lacquer itself having gotten scuffed. The clear bit is made cloudy by the irregular surface. Unfortunately, just like when you scratch up a hardwood floor or the pain on a car, this may mean that you need to remove the lacquer down to where the scuff starts and reapply it, particularly if the waxing you did has put another layer on over the scuff. My advice is to talk to your teacher. He probably has firsthand knowledge of good lacquers and where to find them in your area.
Mold can also develop on the equipment, but generally on the inside and after storage. For that, scrubbing with rubbing alcohol is fairly effective and there are fungicides sold for general athletic equipment that can eradicate the usual offenders.