What does it mean when Mr. White was "amiably desirous of preventing his son from seeing it"?

This quote from "The Monkey's Paw" has a couple of meanings. First, it refers to the chess game that Mr. White is playing with his son, Herbert. You will notice from the paragraph above that Mr. White plays a move with his king that he later realizes gives his son an advantage. This is the "fatal mistake" to which the narrator refers.
Looking deeper, however, we can see that this is an example of foreshadowing. When Mr. White says that he wants to prevent his son "from seeing it," for instance, the "it" might well refer to the monkey's paw. As we see later in the story, it is Herbert who encourages his father to make the first wish, and, as a result, Herbert loses his life in an accident at work. This line, therefore, foreshadows that tragic moment.