Beat flour, poppyseed filling, 2 1/2 cups confectioners' sugar, butter, eggs, lemon juice, baking powder, and lemon peel in a large bowl with an electric mixer set on medium speed until the batter is smooth and thoroughly combined, about 2 minutes.
Pour the batter into an ungreased 10-inch tube pan.
Bake in the preheated oven until the cake is lightly browned and a toothpick inserted into the middle of the cake comes out clean, about 1 hour and 15 minutes.
Cool the cake thoroughly.
Beat 1 cup confectioners' sugar with 1 tablespoon milk with an electric mixer on medium speed for 2 minutes; drizzle frosting on cooled cake.