It sounds as though your potatoes are infected with potato scab, a fungal disease that persists in the soil over long periods. Infection occurs through wounds and through the pores in the tuber skins when the tubers are young and growing rapidly. Potato scab is most severe in warm, dry soil that's slightly alkaline. If your soil pH is above 5.5, lower the pH with powdered sulfur (available at your local garden center). The sulfur makes weak sulfuric acid, which lowers the soil's pH level. Here are other ways to deal with scab. Rotate your potato patch so each place is used for potatoes only once every 3-4 years. Plant resistant varieties; several have high scab resistance.