How can one avoid the negative impact of a husband's parents' disapproval on the marital relationship, and is there a specific supplication for that?

This hatred may be temporary and can disappear with your good treatment and by showing kindness. You must present your best manners to your husband's family, especially his parents, and care for his mother as if she were your own. Turn to Allah with supplication to grant you success in living with them kindly and softening their hearts, for success is in His hands, glory be to Him.