Make sure you keep all wheel and axle bearings very well lubricated. The wheels and axle should spin very free.
Lubricate the chain well after every race. Make sure and apply your chain lube to the inside surface of the chain rollers, where they contact the sprocket teeth.
Decrease rear tire pressure
Move wheels inward on rear axle
Stiffen rear of frame
Decrease front tire pressure
Move wheels inward on spindles
A number of organizations don't have a minimum weight for the kid kart class. This is a significant opportunity. When lightening the kart first concentrate on rotating parts then fixed parts. Remember to be sensible and don't compromise the integrity or safety of the kart. The rotating improvements also help, even if you do run a minimum weight restriction.
Some things to try are;
Aluminum axle
Synthetic axle sprocket
Lightweight sprocket hub
Lightened brake rotor
Lighter attachment bolts and nuts for sprocket and hub
Nerf bars should be loosely attached to the frame. You don't want the nerf bars stiffening the chassis. They can be used as a tuning aid if desired.