Would things have turned out better if Mr. and Mrs. White had phrased the second wish more carefully in the story "The Monkey's Paw"?

No, the monkey's paw was an instrument of dark magic, designed to bring misery to any greedy human who dared try to achieve personal gain through its use. 

It is clear from the previoius owners, both the first owner, whose third wish was for death and Sergeant Major Morris that the act of wishing in itself brings with it severe and tragic consequences.

The lesson that should have been clear to the Whites after hearing the stories from Sergeant Major Morris, is that wishing will bring more harm than good. 

The Whites could not have gained anything but misery and sorrow by using the monkey's paw that was its intended goal.