Jan, the only Derbyshire cake recipe I've found so far is this one in Helen Edden's County Recipes of Old England (1927)
1/2lb moist sugar
1/4 pint milk
Rub the butter into the flour, then add the sugar and currants. Mix with the egg and the milk, form into a paste and roll it out thin, and cut into little round cakes with a cutter. Lay them on a buttered baking sheet, and bake them slowly for about 10 minutes.
In Neil Heaton's 1951 anthology, Traditional Recipes of The British Isles, he gives largely the same recipe but advises an oven temperature of 400 degrees F.