1. Heat oven to 350 degrees . Butter and flour a 9 x 9-inch baking pan. In a bowl, mix 1 cup of the flour, the cornmeal, baking powder and salt. In a separate bowl, whisk sugar, milk, butter and eggs. Fold dry mixture into wet mixture until just combined.
2. Toss blackberries with remaining 1 tbsp flour. Gently fold into batter. Transfer to baking pan, using a spatula to smooth the top. Sprinkle with turbinado sugar, if using.
3. Bake at 350 degrees for 40 to 45 minutes, until a toothpick inserted in center of cake comes out clean. Place on a wire rack and immediately run a paring knife around edge of cake. Cool 20 minutes.
4. Whisk heavy cream and honey in a bowl until stiff peaks form. Serve cake warm or at room temperature with whipped cream.