In the first season of True Detective, as much as I know the reason for the scars on Childress' face is not mentioned.
So why is his face scarred? Is it mentioned clearly in any part or any scene of the series?
It's not clear or directly mentioned but when Errol says
"Do you wanna know what they did to me? What I will do to all the sons and daughters of man."
he partly explains it himself (spoiler alert):
While luring Rust through Carcosa, Errol provides the following information: “Do you wanna know what they did to me? What I will do to all the sons and daughters of Man.” He speaks, here, of the scars upon his face: something happened to him in his youth that gave him these scars, and we can bet that it was a part of the rituals the Tuttle clan engaged in. Thus, Errol explains his actions to us: he is carrying out what was done to him, in his youth, or is at least reenacting some form of traumatic experience. It is clear that Errol was not entirely accepted and beloved by his elders. The lips on the body of Errol’s father are sewn shut, suggesting some deep pathological animosity: the Law of the Father, silenced. Errol experienced something horrifying as a child, and his mission – as well as his means of controlling that experience – is to visit that experience upon new victims.