Applesauce Spice Cake

1 1/2 cups granulated sugar

2 teaspoons ground cinnamon

1 teaspoon baking powder

1 teaspoon baking soda

1 teaspoon salt

1 1/3 cups applesauce

1/3 cup oil

2 eggs

3/4 cup chopped peeled apple (1 medium)

1/2 cup raisins, if desired

1/4 cup butter (do not substitute margarine)

4 cups powdered sugar

1/4 teaspoon ground cinnamon

1 teaspoon vanilla

6 to 8 tablespoons apple juice or milk