The son had already been dead for a while and was beginning to decompose. He knew that it was bad to tamper with fate, because wishing for 200 pounds in money was probably what caused the son to be killed in the machinery.
Mr. White also probably felt a huge sense of guilt and dred at the possibility of facing his son after wishing for the money. Since the money came from the death compensation from the company the son worked for, the father would have to explain it to the son.
Finally, Mr. White was afraid of the curse of the Monkey's Paw and would rather live with the result of one bad wish than the result of more than one bad wish.