The son had been in a machine accident. Possibly his image was marred beyond recognition. The father realized that his son would look like a monster. This would be more than he and wife could have handled.
In fact, the father does not desire to wish for his son to be alive again, but his wife is insistent:
He tries to tell her that since he was mangled in the machinery and had been buried for a week, it would not be a wise wish. But she insists.
This is reason the father made his third wish. He could not bear to look on his son who had been mangled in a machinery accident. The father knew that his wife could not handle what sight she was about to see, had the father not made a third wish.
Mr. White is afraid of what his son must have turned out to be. He doesn't want his wife to see their son who has been in a machine accident. He thinks that they wouldn't be able to deal with it.
"He has been dead ten days, and besides he--I would not tell you else, but--I could only recognize him by his clothing. If he was too terrible for you to see then, how now?"