Preheat your oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
In a large bowl, beat the softened butter and granulated sugar together with an electric mixer until light and fluffy, about 3-4 minutes.
Add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract and lemon zest until fully combined.
Gradually add the flour mixture to the butter mixture, alternating with the sour cream. Begin and end with the flour mixture, mixing until just combined.
Gently fold in the fresh blackberries and lemon juice, being careful not to break them apart too much.
Pour the batter into the prepared cake pan and smooth the top. Scatter a few extra blackberries over the top.
Bake in the preheated oven for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
Once baked, allow the cake to cool in the pan for 10 minutes before transferring to a wire rack to cool completely. Dust with powdered sugar before serving.
Notes
Serve warm or at room temperature. Dust with powdered sugar just before serving.