Preheat oven to 350°.
Spoon applesauce onto several layers of heavy-duty paper towels; spread to 1/2-inch thickness. Cover with additional paper towels; let stand 5 minutes. Scrape applesauce into a bowl using a rubber spatula.
Combine sugar and butter in a large bowl; beat with a mixer at medium speed until light and fluffy (about 2 minutes). Add applesauce and banana, beating until blended. Add eggs, 1 at a time, beating well after each addition. Beat in coffee and vanilla.
Lightly spoon flour into a dry measuring cup; level with a knife. Combine flour and remaining ingredients except powdered sugar, stirring well with a whisk. Add flour mixture to egg mixture; stir just until combined. Scrape batter into an 8-inch baking pan lined with parchment paper, spreading evenly. Bake at 350° for 40 minutes or until a wooden pick inserted in center comes out clean. Cool in pan on rack. Sprinkle with powdered sugar, if desired.