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.