Hand grips are super important to keep maintained. If your child's bike's hand grips are punctured at the ends, they should be replaced immediately because in the event of a fall, the exposed metal can cause serious injury.
SRAM provides an exceptional quality hand grip that is highly popular as a replacement for any bike with industry standard 22.2mm diameter handlebars. (This covers 99% of all bikes, don't worry) These make a great upgrade, most kids bikes come with inferior quality hand grips.
Installation tip: remove the old grips by cutting them off, no need to do extra work. To install the new grips, make sure the surface of the handlebars is free of residue or adhesive. Lightly moisten the handlebar with a Windex type of cleaner (not too much!) and the grips will slide onto the handlebars with a minimum of force. Let the bike sit in a well ventilated area for 24 hours. The Windex will evaporate and the grips will end up being very secure. If too much Windex is used, the dry time will be longer, so use just enough to slip the grips onto the bars.