Preheat the oven to 350℉ (180℃).
Make the espresso swirl. In a small bowl whisk together the ⅓ cup packed brown sugar, 1 tablespoon instant espresso powder and ½ teaspoon cinnamon. Set aside.
Make the pumpkin cake batter. In a large bowl whisk together the flour, baking powder, baking soda, spices and salt. In a separate large bowl whisk together the oil, brown sugar, granulated sugar, vanilla, eggs, pumpkin, and sour cream. Keep whisking until there are no lumps of sugar. Gently, whisk the flour mixture into the wet ingredients until no lumps remain
Assemble the cake. Spray the bundt pan with non-stick baking spray, ensuring that all the crevices are coated. Alternatively, grease and flour the pan or brush with pan release. Spoon about ⅓ of the cake batter (about 2 cups) into the pan. Sprinkle ½ of the espresso filling on top. Then spoon another ⅓ of cake batter on top - so that the espresso filling is covered. Sprinkle the remaining ½ of espresso filling on top. Lastly, spoon the remaining cake batter on top.
Bake the cake in the middle of the preheated oven for 55-70 minutes. When finished, an inserted toothpick should come out clean or with a few moist cake crumbs. There may be some of the brown sugar espresso filling on the toothpick but there should be no gooey cake batter. The cake should be starting to pull away from the sides of the pan.
Cool the cake in the pan for about 20-30 minutes, then very carefully invert the cake onto a cooling rack to continue cooling.