Preheat the oven to 375°F (190°C). Butter and flour a 9-by-5-inch loaf pan or similar cake pan.
Whisk the eggs and powdered sugar together in a large bowl until smooth. Add the sifted flour and baking powder, then mix in the melted butter.
Peel and core the apples, then grate them using the large holes of a box grater. Fold the grated apples into the batter.
Add the warmed salted caramel sauce and cinnamon, then fold gently until evenly combined.
Pour the batter into the prepared pan and smooth the top. Bake for about 45 minutes, or until a knife or toothpick inserted in the center comes out clean.
Cool the cake in the pan for 10 minutes, then turn it out onto a rack and let it cool before slicing.