I think Hakari will be some sort of more rebellious version of Gojo. So basically he will be like Gojo, uh oh fuck the higher-ups but like more intensively.
When it comes to Yuuji running away, that depends. If Yuta and Hakari will be on Yuuji's side the higher-ups can do shit. There is a huge chance Yuta is the strongest sorcerer at the moment while Gojo is out of commission.