An objects mass is not affected by gravity. No matter where an object is, it will have the same mass. How much it weighs will change, because gravity will change.
No, the mass of the given object will be same on the Earth as well as on the Moon.
Explanation:
The term "mass" is estimation of a given object or a system of objects. It is a totally independent quantity. Thus, its magnitude will not vary while carrying it from one place to another.