Haymitch is a former victor of the Hunger Games, and as a result carries the trauma and guilt of having survived while others died. He is often dismissive of the Capitol and the Games, and his cynicism and world-weariness make him a difficult person to get close to. However, he does develop a connection with Katniss and Peeta, who both remind him of his own younger self.
Haymitch shows Katniss and Peeta that he cares for them, but does so in a way that is typically blunt and indirect. He is the one who teaches them about the strategies and tactics they can use to survive the Hunger Games and also offers them support and guidance when they need it most. He also shows them kindness and compassion, such as when he gives Katniss a pin of a mockingjay, a symbol of hope in the rebellion against the Capitol.
Haymitch is a complex and imperfect character, but he does ultimately show that he cares for Katniss and Peeta in his own way. He is a survivor who has learned the hard way how to look out for himself, but he is also a compassionate and caring person who is willing to fight for what is right.