Lyons is Troy's son from a previous relationship. Troy had Lyons when he was still homeless and squatting in a shack by the river. Lyons's mother moved on to another man while Troy was in prison, so Troy has never been much of a father to Lyons. Pretty much all Troy contributes to Lyons's life is the occasional ten dollars that Lyons comes to bum on Troy's payday. (And Troy even grumbles about that).
Interestingly enough, Lyons doesn't seem particularly bitter about any of this. He just seems to accept things as they come. Even though Troy won't even take the time to come see Lyons play music (the real pride of Lyons's life), Lyons still seems to respect his father. The almost easy going relationship between Lyons and his father is starkly different from the tense rivalry between Cory and Troy.