Preheat the oven to 350 degrees. Spray an 8 x 8 baking dish with cooking spray.
In a large bowl mix together the flour, baking powder, ginger, salt, and nutmeg. Set aside.
In a medium bowl cream together the butter and sugar. Add in the eggs and mix well. Stir in the vanilla.
Add half the flour to the butter mixture and mix until combined. Mix in the sour cream. Then add the remaining flour to the butter mixture. Mix until well combined.
Spread the batter into the prepared baking dish.
If the blackberries are really big cut them in half. Spread the raspberries and blackberries over top of the cake batter in the pan.
In a small bowl combine the brown sugar, flour, oats, ginger, and salt. Add the butter and mix with your fingers until a crumble forms.
Sprinkle over top of the berries.
Bake the cake for 50 minutes to 1 hour or until a toothpick inserted into the center comes out clean.
Place the cake on a wire rack and cool completely.
Whisk the powdered sugar and lemon juice together. Drizzle over top of the cooled cake.