There are several problems Robin Hood is facing now. First, since he welcomed all who came without even asking enough questions to get to know them. Even though the band was expanding, the decline in discipline and vigilance has already been a big threat for the band. Food shortage was a result of the increasing size of the band. Since the travelers, whom the band used to rob from, were now giving the forest a wide berth, which made the source of fund a big issue. On the contrast, "the sheriffs was growing stronger and becoming better organized".
I don't think Robin Hood needs a new mission. The pro for securing King Richard was obvious. Since Robin Hood's two major rivals were sheriff and his admiration. Once the old king was taken down, his government structure need to be reorganized, which means Robin Hood and his band got a chance to get rid of the sheriff. However, the con seems more likely to happen. Since Prince John has spies everywhere, and he was vindictiveness, which means the failure of recurring King Richard could lead the band to death.