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Preheat oven to 350 degrees F. Lightly grease a 9-inch round cake pan. For easier removal of cake from the pan, line the bottom of the pan with parchment paper and lightly grease parchment.
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Whisk together flour, baking powder, baking soda, and salt in a large bowl. Add the apricot pieces and whisk to combine.
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Whisk together eggs, buttermilk, 1/2 cup sugar, and butter in a separate bowl.
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Pour liquid ingredients into dry ingredients and stir to until just combined.
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Spread batter evenly in prepared pan.
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Dollop jam by teaspoonfuls over batter and swirl with a knife.
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Sprinkle walnuts on top of batter. Sprinkle remaining 1 teaspoon sugar on top of walnuts.
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Bake for 35 to 40 minutes, until a toothpick or cake tester inserted in the center comes out clean.
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Cool cake in the pan for 7 minutes on a wire rack. Remove cake from pan and cool completely on a wire rack.