Richard was using his status as the crown prince to look for prince William. Although he did not care about the life and death of Prince William as the crown prince this was his duty.
Richard also had in mind that he will leave the seat of the crown prince after this. The King favored Williams and it might put his life in danger.
Growing up Richard had never had his eyes for the throne, not even when he first found out he was the son of the king.
While he was investigating what happened to William he learn of the news that Prince William has escaped with a group of soldiers a few days back. Since only a small group of people knew about the real reason why the group of men set out to Avalo, Richard was unable to find out their destination.
Since prince Williams was no longer in Pontus, Richard decided to follow Jeffery and the twenty.
"I will go with you," Richard told Jeffery who was arranging to leave the camp.