Preheat oven to 425 degrees. Oil a 9- or 10-inch cast-iron skillet and put into oven to heat while the oven preheats. (Note: If you do not have a cast-iron skillet, you can bake this in a 9-inch square pan instead, just skip the preheating.)
In a large bowl, whisk together the cornmeal, flour, baking powder, salt and 3 tablespoons of sugar.
In a separate bowl, whisk together the eggs, buttermilk and butter; add to the dry ingredients, stirring just until combined.
(Carefully) remove preheated skillet from oven; pour batter into pan. Quickly distribute blackberries over the top of the cornbread, gently pushing down into the batter. Sprinkle with the remaining tablespoon of sugar. Bake for 12 minutes or until lightly browned.