The doughnut traditionally made with mashed potato and popularized by Spudnuts shops is a spudnut.
A spudnut is a doughnut made partly with potato or potato flour. The potato contributes moisture and starch, producing a distinctive texture that can be soft and slightly dense while still allowing the dough to be fried like a conventional doughnut.
The Spudnut Shop enterprise was associated with brothers Bob and Al Pelton, who promoted a potato-based doughnut mix in the United States during the twentieth century. Independent shops using the name appeared in many communities, although they were not all part of one continuously controlled national chain.
Spudnuts are often confused with ordinary cake doughnuts because both can have a dense crumb. The defining feature is the potato ingredient, not merely the absence of yeast or the shape of the pastry.