While children can be proficient mountaineers and well trained, there are things they likely cannot do and some additional risks they face. For example, children are more susceptible to hypothermia. Most children are not capable of performing first aid and lack the physical strength to dig out someone trapped in an avalanche. If a single adult is climbing with a child, they should consider themselves as essentially climbing alone.
A child climbing alone with a single adult also faces a unique risk in that if the adult falls into a crevasse, the odds of an unanchored child catching such a fall are pretty low (the exact odds depend on how the adult falls, the steepness of the slope and how icy things are). The adult and child will both likely end up wedged at the bottom of the crevasse. This must be prevented at all costs. Basically this means that whenever there is a chance the adult could fall (i.e., always on the glacier), there needs to be adequate protection. You might be able to get by with a 4th class running belay placing pickets, screws or deadman as you go, but you would probably want to go very slow and just set 5th class belays and go a rope length at a time. With a solid anchor a child can catch basically any fall.
It is worth noting that if the child some how manages to catch the fall such they both do not end up in the crevasse, then things are not too bad as long as the child is capable of setting an anchor while holding the fall and knows how to build a hauling system and appropriately protect the lip (etc). If the child knows the skills, their size should not prevent them from getting the adult out. There of course is the issue of first aid and hypothermia once the child gets the adult out.
If the child is not highly skilled and trained, then one adult and one child over any technical terrain is ridiculously unsafe. Two adults and one untrained child has a lot of similar risks are two adults, but if one adult falls, they will likely pull the child off leaving the remaining adult having to catch an adult and a child (good luck with that).