Whisk together the yogurt and 2 tablespoons of the sugar in a medium bowl.
In a separate bowl, combine the blackberries and the remaining 2 tablespoons of sugar and mash with a fork or potato masher until coarsely crushed.
Layer the yogurt and blackberry mixtures (starting with the yogurt mixture), dividing evenly among eight 4-ounce paper cups or 4-ounce ice-pop molds. Drag a toothpick or skewer through the mixtures to create a swirl effect. Insert ice-pop sticks and freeze until firm, at least 3 hours and up to 4 days. Remove from the freezer 10 minutes before serving to soften slightly.