Swans are grazers, meaning they eat aquatic vegetation and grasses. So, unlike predatory birds as owls, hawks, etc., they do not kill and fly away with prey. So, we’re really not sure what they would be carrying while they are flying.
Furthermore, although mother swans carry their young on their backs beneath their wings, they only carry the young while the female is floating on water, not during flight. The carrying of young on the back only lasts for a couple of weeks while the cygnets are extremely small and weigh only ounces. We’re not sure there is an appropriate answer to the question of how much they can carry during flight as they typically do not carry anything while flying. The Regal Swan