Preheat the oven to 350ºF. Grease a 9-inch round cake pan and dust with flour. Combine the flour, cornmeal, baking powder, and salt in a medium bowl.
Combine the butter and sugar in a large mixing bowl and cream with an electric mixer on medium-high speed until fluffy, about 3 minutes, scraping down the sides of the bowl once or twice as necessary. With the mixer on medium-low speed, add the eggs, one at a time, scraping down the sides of the bowl after each addition.
Turn the mixer to low speed and add 1/2 of the flour mixture. Stir in the milk. Stir in the remaining flour mixture until just combined. Stir in the cheddar cheese and apple.
Scrape the batter into the prepared pan and smooth the top with a rubber spatula. Bake until the cake is golden brown and a toothpick inserted into the center comes out clean, about 35 minutes. Let the cake cool in the pan for about 10 minutes, invert it onto a wire rack, and then turn it right side up on a rack to cool completely.
Store uneaten cake in a cake keeper or wrap in plastic and store at room temperature for up to 2 days.