Preheat the oven to 350 ⁰F. Butter and flour an 8 1/2-inch-by-4 1/2-inch loaf pan. In a medium bowl, whisk together dry ingredients: flour, baking powder, cinnamon, salt, nutmeg and cloves. In a large bowl, whisk together the wet ingredients: oil, applesauce, sugars, eggs and apple.
Then fold the dry ingredients into the wet ingredients until just combined. Transfer to the prepared pan and bake until a toothpick inserted into the center comes out clean, 50 to 60 minutes. Afterwards, transfer to a rack to cool slightly, then remove the cake from the pan and let it cool completely.
When the cake has cooled, prepare the glaze: Bring the apple cider to a boil in a small saucepan over medium-high heat. Continue to cook the cider until it has reduced to 1/3 cup, about 10 minutes. Add the butter, confectioners’ sugar and salt. Reduce the heat to maintain a simmer and continue to cook the mixture, whisking often, until it has become syrupy and thickened slightly, about 4 minutes. Set aside to cool until it thickens slightly.