More than a month after Hanjin Shipping Co. sought bankruptcy protection in South Korea and in courts around the world, customers are still fighting with the company over how to retrieve their goods.

Ashley Furniture Industries Inc. said it has been left on the hook for cleaning up the logistical mess in the wake of Hanjin’s bankruptcy and is asking a U.S. judge to allow it to withhold damages from fees it owes to Hanjin. The South Korean shipping company, however, is refusing to release some of Ashley Furniture’s cargo until...