Mr. White does not fully believe the story of the monkey's paw, but since he is susceptible to suggestion, he allows himself to be drawn into the idea that the paw may hold supernatural powers. Despite this, he takes the paw from his friend Morris as a gesture of solidarity; he knows that Morris has been through a lot of tragedy, and he wishes to take some of that away. Morris, for his part, is well-aware of the paw's capabilities, and hopes that White will remain skeptical and not use it.
"I don't know what to wish for, and that's a fact," he said slowly. "It seems to me I've got all I want."
"If you only cleared the house, you'd be quite happy, wouldn't you?" said Herbert, with his hand on his shoulder. "Well, wish for two hundred pounds, then; that'll just do it."
His father, smiling shamefacedly at his own credulity, held up the talisman...
(Jacobs, "The Monkey's Paw," gaslight.mtroyal.ca)
As seen in the prior excerpt, White is hesitant and only wishes because, in his mind, no harm can come of it. He cannot fully understand the consequences of "meddling with fate," as Morris puts it; instead, he thinks that the wish will either have no effect, or he will gain the money. Of course, his hopes are dashed, and he is forced to face the truth. His first wish stems entirely from his skepticism and his general wishes to keep his wife and son happy; they are cheerful to go along with the farce, and none of them believe that it can actually have a real effect.