Preheat oven to 350 degrees. Grease a 10-inch bundt cake pan.
In a mixing bowl, combine cake mix, pudding, milk, eggs, zest, and fresh lemon juice, for 2 minutes with electric mixer. Pour batter into prepared pan and bake 40-45 minutes or until toothpick inserted comes out clean.
Let cake cool in pan on a wire rack for 10 minutes. Invert cake onto a serving plate. Let cake cool completely before glazing.
To make the glaze, combine powdered sugar, vanilla, and 1 tbsp lemon juice. Use a spoon to combine, then spoon over cooled cake.