From PlayStation Universe: “As is customary in the Tales of franchise, many bosses feature secret missions that yield extra rewards for completing them. Tales of Vesperia: Definitive Edition is no different. These items arent necessarily crucial, but completionists like me have a mighty need to find everything. From a Trophy standpoint, completing these missions is a must for two counts. Each mission has its own Trophy, of course. A separate Trophy requires obtaining all items in the game. Some of those items are unique to these secret missions.”
Source: N4G PC Tales of Vesperia – Secret Missions Guide