I think Wriothesley is in general more open about showing that he cares for Lyney, but he does it in a VERY roundabout way. If he's too open or honest, Lyney might hiss at him and hurry off after all. Lyney very much does not want to deal with Wriothesley in the beginning and he's content disliking him.

If Wriothesley were to find out that Lyney is sick, maybe because Lynette came to a meeting in his stead, he'd definitely send her back with a tin of soothing tea to get him back into his feet. Once they're closer, maybe he'd visit him and offer to look after him a little. All the while chiding him to look after himself better and to take breaks.

If Lyney's injured... I love thinking about the possibility of Lyney going to Wriothesley rather than his home. He doesn't want to worry Lynette or Freminet after all. He doesn't question why it feels easier to go to Wriothesley, and Wriothesley wouldn't question it either because... They just don't do things like that. He'd tell Lyney to sit his ass down and mutter under his breath while he cleaned and dressed his wounds, calling him reckless and whatnot... Wriothesley definitely cares, way more than what he should and than he can hide from Lyney.