slipping belts can be caused by "insufficient" throttle
there's always a range of rpms where the clutch is just partly engaged and slips,
you need to get over that transition (clutch starting to engage -> clutch fully engaged)
as quickly as possible in order to keep the belt as vital as possible.
The hot belt then heats the variator which softens the roller coating and
after a while it looks like what you hold in your hands.
You don't happen to have limited the throttle (or pedal for that matter) in order to slow your kid down considerably [likely to seduce mom's demand for 'not too fast!'], have you?
in case you did.. THAT is the direct result.
Another cause would be too steep of an inclination to climb
and/or too heavy of a weight to carry
(say increeased rear wheel diameter and running at or beyond the loading capacity...)
If your boy packs six or seven of his best friends inside occasionally you can be rather certain it's been overloaded
You can likely clean the variator plate with acetone..
just allow it to soak in for a few hours and take an old toothbrush for the then sticky residue.
While at it, clean the clutch side as well.
it might look almost as bad (maybe glazed by now, lightly sand it or if that fails replace the shoes)
Take care that the gear ratio (rear wheel size as well as sprocket ratio) is okay,
and that the throttle isn't limited to get the motor to rev up to it's full potential.
And if you stay below the max capacity the now fully restored CVT (clutch, belt and variator)
should be able to move your son and his kart for a good amount of time reliably
'sid