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The way Hawks looks at Tokoyami reminds me of just how fond he is of the 1-A student. To me it shows that he didn't expect help from anyone, and that when Tokoyami comes in with Jirou to save him, again, Hawks realizes just how much Tokoyami cares for him, and is willing to risk is own life for his mentor'a life. He admits to not having the raw power Endeavor has, but he knows that Hawks knows that he has the ability to keep up and have better coordinated attacks. Why? Because Hawks doesn't have to worry about getting burned by Endeavor if they were to mess up, and because Hawks knows what Tokoyami is capable of. Hawks never wanted to become a teacher or train the heros of the future, but he was willing to show Tokoyami how versatile his quirk really is. Thus another reason he was willing to continue his studies and work with Hawks even after his first interactions with Hawks was nothing more than giving him intel.
Hawks admits to hardening his heart and the will of forgetting any relations he has had in the past, to continue moving forward and creating a better future. Though I don't think he could ever shove Tokoyami away, after everything they have been through, Tokoyami never gave up on Hawks. Even when he admitted to killing Twice, Tokoyami is still willing to fight along side Hawks and defeat the evil that corrupts the world.
Also here is colored version. I absolutely love their dynamic of student/mentor.