Begin by preheating your oven to 325°F (165°C). Prepare two 8-inch round cake pans by greasing and flouring them. If you intend to split the cake layers, use two pans, but if not, three pans will work as well. Line the base of each pan with a circle of parchment or wax paper for easy removal.
In a mixing bowl, blend the dry ingredients: the cake mix, flour, and pudding mix. Stir to combine thoroughly. Add the eggs, sour cream, water, oil, and vanilla extract to the dry mix. Blend on low speed, gradually increasing to medium speed for about one minute, scraping down the sides of the bowl as needed. Continue mixing for another minute to ensure a smooth batter.
Evenly divide the batter between the prepared pans.
Bake at 325°F (165°C) for around 30 minutes, or until a toothpick inserted into the center comes out mostly clean with just a few crumbs. The cake should also bounce back when pressed gently. If using three pans, reduce baking time to 22-25 minutes.