Preheat the oven to 160ºC and line a 23 cm (9 inches) square baking tin with nonstick baking paper.
Prepare the apples: Peel, core, and chop the apples. Toss them in a bowl with the sugar and cinnamon, then set aside.
Cream the butter and sugar: Use an electric hand whisk to beat the butter and caster sugar until very light and fluffy, aiming for the consistency of thick mayonnaise.
Add the eggs: Add the eggs one at a time, mixing well after each addition. The mixture may curdle slightly, but it will come together once the flour is added.
Combine dry ingredients: Add the plain flour, baking powder, and cinnamon to the mixture. If using self-raising flour, skip the baking powder. Gently fold everything together with a wooden spoon.
Assemble the cake: Spread the batter evenly into the prepared baking tin. Distribute the chopped apples evenly over the top.
Bake: Place the cake in the preheated oven and bake for 35-40 minutes, or until a skewer inserted into the center comes out with a few moist crumbs.
Finishing touch: If using demerara sugar, sprinkle a tablespoon over the hot apple cake after it comes out of the oven to add a delightful crunch.