Preheat oven to 350°F and prepare a 9" round cake pan with parchment paper or grease it.
Core, peel, and chop apples into ½" cubes, tossing them in lemon juice to prevent browning.
Whisk together flour, baking powder, salt, and cinnamon in a small bowl.
Cream butter and sugar in a stand mixer until light and fluffy, then add eggs one at a time, mixing well.
Mix in rum, then gradually add the flour mixture until just combined. Fold in the chopped apples.
Pour the batter into the prepared pan, smooth the top, and sprinkle with sugar.
Bake for 40-50 minutes, let cool, and optionally dust with confectioners' sugar.