What mistake does the man make in building his 2nd fire? What happens as a result?

I think you are referring to the third fire. The mistake that the man makes in the second half of the story, is that he builds his fire with his last match under a tree. While the fire initially catches hold, it also melts the snow on the branches above the fire. Once the snow melts and falls on the fire and puts that fire out, the man has no more choices. He can no longer build a fire; without it he will freeze to death.