This is really good, the applesauce makes this cake really moist! My family prefers a sweet tasting cake, but you can reduce the white sugar amount slightly if desired. Use sweetened or unsweetened applesauce.
Set oven to 350 degrees (set oven rack to second-lowest position).
Grease an 11 x 7-inch baking pan.
In a bowl cream the butter with both sugars until creamy.
Add in eggs and vanilla; beat until well mixed.
Add in applesauce; mix until combined.
In a small bowl mix together flour, baking powder, baking soda, salt, cinnamon, nutmeg and allspice; add to the creamed mixture; mix just until combined.
Mix in the apples and raisins (if using).
Transfer to prepared baking dish.
Bake for about 35 minutes, or until the cake tests done.