For the cake: Position a rack to the center of the oven and preheat the oven to 350 degrees F. Spray an 8-inch square, light-colored metal baking pan with nonstick cooking spray, and then line the pan with an 8-inch wide strip of parchment paper, leaving several inches of overhang on the opposite sides of the pan.
In a medium bowl, sift together the flour, baking powder, salt and baking soda. In a separate medium bowl, using a handheld mixer, beat the egg whites to stiff peaks.
In another medium bowl, whisk together the egg yolks, honey, applesauce, oil, lemon juice, vanilla extract and lemon zest. Pour the wet ingredients into the flour, and fold gently to blend. Add about one-third of the whipped egg whites to the batter, and stir to blend well. Add the remaining whipped whites, and fold gently just until no visible clumps of the egg white remain.
Remove the berries from the freezer. Scrape half of the batter into the prepared pan and smooth evenly. Sprinkle half of the frozen berries over the batter. Pour and smooth the remaining batter, and then scatter with the remaining berries. Bake until the cake is deep golden brown and a toothpick inserted in the center comes out with a few moist crumbs, 50 to 55 minutes.
Cool for 10 minutes in the pan set on a wire rack.
For the glaze: Whisk together the honey and lemon juice in a small bowl.
Spoon the glaze over the warm cake, letting it absorb before adding more. Cool the cake completely, and then slice and serve.
This cake is even better the next day; wrap tightly in plastic wrap overnight before serving.