My potatoes have got a scab. These appear to been caused by minute white 'worms.'
Certified GKH Gardening Expert
There is a disease called potato scab which is caused by bacteria, but insect or other invertebrate damage could cause the wounds that allow the bacteria in:
If those pictures don't match what your potatoes have, the "worms" you see might be nematodes, which can cause several diseases on potatoes. Some of the diseases cause a scabby appearance.