Preheat your oven to 350°F (175°C). Grease and line an 8x8-inch baking pan with parchment paper, leaving an overhang for easy removal. In a mixing bowl, whisk together the melted butter, sugar, eggs, and vanilla extract until smooth. Add in the flour, cocoa powder, salt, and baking powder. Stir until well combined, forming a thick brownie batter.
In a separate bowl, beat the softened cream cheese and sugar until smooth and creamy. Add the egg and vanilla extract, and mix until fully incorporated. The mixture should be smooth with no lumps.
Spread half of the brownie batter evenly into the prepared pan. Then, spoon the cheesecake mixture on top of the brownie batter. Gently spread it into an even layer. Drop spoonfuls of the remaining brownie batter on top of the cheesecake layer and swirl it together with a knife or toothpick to create a marbled effect.
In a small bowl, toss the blackberries with sugar. Gently place the sugared blackberries on top of the cheesecake layer, pressing them slightly into the batter. This will allow them to bake into the brownies and infuse their flavor into every bite.
Bake the brownies in the preheated oven for 40-45 minutes, or until a toothpick inserted into the center comes out with just a few moist crumbs. Let the brownies cool completely in the pan on a wire rack. Once cool, lift them out using the parchment paper and cut into squares.