newtonian mechanics - Verify conservation of energy in gravitational field

Remember we can verify conversation of energy in gravitational field when we take $g$ to be constant When we leave body from $H$ At a point $h$ As potential energy is $mgH -mgh$ Kinetic energy $=mv^2/2= mgh$ as $v=\sqrt{2gh}$ Thus total mechanical energy is $mgH$

Similarly can we verify the conservation of energy for variable gravitational attraction $g=GM/R^2$?