Preheat an oven to 350° F and line one 8" round cake pan with parchment paper.
In the bowl of a mixer fitted with a paddle combine the softened butter, sugar and lemon zest. Begin creaming the butter and sugar, scraping down the sides of the bowl as needed, for 4 minutes. Or until light and fluffy. Do not rush this process.
In a separate bowl combine the flour, baking powder, baking soda, and salt. Set aside. Measure out the milk, add in the sour cream and stir until fully combined. Set aside.
Once the butter and sugar have become light and fluffy, add in the eggs, one at a time, mixing well after each addition. Beat in the vanilla extract then the lemon juice.
Add ⅓ of the dry ingredients to the butter mixture, mix on low just until combined. Pour in ⅓ of the milk mixture, mix on low until just combined. Continue this process until all of the flour and milk mixtures have been incorporated. Scraping down the sides of the bowl as needed.
Fold in the blackberries.
Pour into prepared pan. Bake for 40-50 minutes. To check if the cake is done, insert a toothpick in the center of the cake, if the toothpick comes out with wet batter on it, bake longer. If the toothpick comes out with moist crumbs on it, remove from the oven and allow to cool for 10 minutes before removing from the pan. Place on a cooling rack and allow to cool completely before icing.
Once the cake has completely cooled, whisk together the powdered sugar and lemon juice. Starting with 1 tablespoon of the lemon juice, add up to ½ tablespoon more if needed to thin the icing out.
Spread evenly over cake. Enjoy!