How do I tune slipping gears?
First got though  this park Tools guide

If you are still having problems, I suspect the derailleur cable. Either replace it (best option as they are cheap) but its worth flicking the outers off the bosses, cleaning and lube as much of the cable as possible.  If this improves things, a new cable may be needed. 

If this does not help you probably need to check the derailleur alignment. unfortunately your derailleur has an integrated hanger so its harder to check. Fortunately, as its only 5 speed, accuracy is not so important. If noticeably bent, you can straighten it using an adjustable spanner, but will have trouble getting it accurate. 

Another possibility is the derailleur is flogged out, although in my experience kids don't change gears enough for this to happen