What is the weight of a 10kg Mass pushed by a 5 N force

As long as it's on Earth, 10kg of mass weighs 98.1 Newtons.

It doesn't matter if it's being pushed, pulled, mashed, heated on the stove, frozen in a snowdrift, painted black, shot from a cannon, hung from the ceiling, sunk in mud, licked by a puppy, or just left sitting in the back of the drawer. None of these has any effect on its weight.

To change its weight, you'd have to either change its mass ... cut a piece off of it or glue another piece onto it ... or change the strength of gravity ... like take it to the moon or to another planet.