Weight is a force. Correctly measured in Newtons, not kilograms. Weight is a product of mass times acceleration. That's Newton's Second Law. Rest-weight is mass times $g$ the gravitational acceleration constant. But if that object accelerated faster than $g$, wouldn't it weigh more than it would when it is not moving?
This question is probably more a matter of terminology than actual physics.
I believe that you imagine a piece of matter falling with an acceleration that is higher than $g$ and if that can count as "more weight". If weight is defined as net force of object downwards towards earth then sure it has "more weight". However "weight" is usually defined as $mg$ meaning that it will have the same weight independently of other forces than gravity giving it a higher acceleration than g.