Headcanon that the first time Kaz manages to hug Jesper, it’s after Colm has passed.
Obviously this s some years in the future — it would have taken Kaz a long time to get to the point of being comfortable with that much contact. So, up until this point, the most Jesper — or most of the Crows — have gotten are pats on the back with his gloves still on, brief pats just to congratulate them after jobs. Which was still an improvement, and Jesper probably thought he was glad he was even close enough to Kaz for him to do that.
But then, Colm passes. Jesper is a grieving mess, to say the least. Colm likely passed more peacefully than his mother, but it’s still losing his last parent.
All of the Crows help him make funeral arrangements and plan to go with him to Novyi Zem to bury his father there. They are all incredibly caring and do their best — especially Wylan, of course — but while they can certainly empathize and listen to him talk about it when he needs to, Kaz is still the only one out of all of them who has lost a father to death.
So, the night before they leave for Novyi Zem, Jesper can’t sleep. He’s sitting up at the kitchen table instead, crying to himself in that tired way you do after you’ve been grieving for a few days and don’t have it in you to sob but the grief still needs to expel itself, so the tears just roll down your face quietly and you stare into space. Kaz still doesn’t sleep much, and so he comes downstairs and finds him there.
He’s sitting comes and sits down next to Jesper — not across from him, next to him — and just is there with him, letting the grief exist in the silence. Jesper is less surprised by this than he used to be, since Kaz has become a bit less frosty over the years amongst the Crows, but he still appreciates it.
What does surprise him is when Kaz suddenly shifts in the chair next to him, and puts on arm around Jesper’s shoulders, squeezing his shoulder with an ungloved hand. It takes Jesper a moment to register it, but when he does he has to force himself not to cry harder, scared it will spook Kaz away or something like that. Kaz knows. Kaz stays as long as he can manage — which still is only a few minutes, but he does it. Then he hugs Jesper tighter for just a moment before letting go, and muttering something about making them both coffee if they were going to be staying up. Jesper smiles through the tears, unable to help it.
They stay up all night after that, not talking much, just staying with each other. They’re still there when the others wake up and get ready to go in the morning.