Position a rack in the center of the oven and preheat the oven to 350 degrees F. Generously butter an 8-inch springform pan (use about 1 tablespoon of butter); line the bottom of the pan with parchment paper. Place the springform on a baking sheet also lined with parchment paper.
In a medium bowl, whisk together the flour, baking powder, salt, cinnamon, cloves, allspice, ginger, and nutmeg.
In a large bowl, whisk together the eggs until they are foamy. Add the sugar and whisk until well blended, about 2 minutes. Whisk in rum and vanilla. Whisk in half the flour and when it is incorporated, whisk in half the melted butter, followed by the rest of the flour and the rest of the butter. Mix gently - so that the resulting batter is smooth and thick. Using a rubber spatula, fold in the apple chunks making sure that each piece is coated with the batter. Scrape the mixture into the springform pan and smooth it out as best as you can with a spatula. The raw cake will look lumpy, almost as if you don't have enough batter.
Place the pan into the oven and bake for 60 to 70 minutes, or until the top of the cake is golden brown and a knife inserted in the center comes out clean. You might see the cake pulling away a bit on at the sides. Transfer to a cooling rack and allow to rest for 15 minutes.
Using a butter knife, carefully run it around the edges of the cake, and carefully remove the sides of the springform, being careful not to pull away any apple bits. Allow the cake to cool to room temperature before removing it from the bottom of the springform (same process as removing the sides). Serve at room temperature. The cake will keep for a few days - best to throw a kitchen towel over it rather than plastic wrap.