Preheat the oven to 350°F. Grease and flour a bundt pan thoroughly.
In a large bowl, cream together the butter and sugar until light and fluffy.
Add the eggs one at a time, mixing well after each addition, then stir in the lemon zest, lemon juice, and vanilla extract.
In a separate bowl, whisk together the flour, baking powder, baking soda, and salt. Add the dry ingredients to the batter in alternating additions with the buttermilk, beginning and ending with the dry mixture, mixing just until combined.
Toss the blackberries with the tablespoon of flour and gently fold them into the batter.
Pour the batter into the prepared pan and smooth the top, then bake for 50–60 minutes, or until a toothpick inserted comes out clean.
Let the cake cool in the pan for 10–15 minutes before turning it out onto a wire rack to cool completely.
Whisk together the powdered sugar, lemon juice, and zest until smooth, then drizzle the glaze over the cooled cake before serving.
Garnish with additional lemon zest, if desired.