<Pedantic mode>
We have a mass of a certain number of Newtons that only becomes a weight in the presence of a particular value of gravity.
Not quite. A mass is usually measured in kilograms (but you could admittedly use grams, tons or pounds if so you wish), but emphatically
not in newtons. Newtons are used for measuring forces, including weight, but certainly not for measuring masses.
So if an individual has a mass of, say, 70 kg, he or she would weigh about 686.5 newtons on the Earth at sea level, i.e. the 70-kg mass of that person would be attracted toward the center of the Earth with a force of 686.5 newtons. On the moon's surface, that person would still have a mass 70 kg, but would weigh only about 113.4 newtons.
</Pedantic mode>