It's hard to tell without seeing the dog but my guess in this scenario would be that he is trying to tell her to back off and may have already tried more subtle messages that were not understood, or may have learned in his previous home that more subtle communication does not work and so taken to giving more explicit signals.
Air snapping is a classic back off message. Because it makes the other party take a step back but without resorting to injury.
I would be...
a) researching what those more subtle signals look like. Check out the ladder of aggression (don't be scared by the title) and look for example videos.
b) talk to your 9 year old about the signals and understanding them. There is a nice illustration of dog body language here: bowwowinsurance.com.au/pet-community/pet-talk/dog-communication-body-language/ The artist (Lili Chin) has also done a cute little book that might be nice for your nine year old to have and read
c) Actively on the look out for your dog given those signals and noting what kinds of scenarios they occur in
d) making sure this dog gets plenty of calm and space. They should always be able to get away from children easily and quickly, to somewhere peaceful.
e) making sure they are always supervised together and I would not be encouraging much interaction at all between them right now. They need time to build up a trusting relationship.
f) Working on using treats to habituate/counter condition the dog to human handling. This dog may be used to children and come from a nice home but few owners actively do this and all should. Accepting human handling does not come naturally to dogs and the more you can help them get used to it, the better for all.
As with any new-to-the-home dog, any interaction or fuss should be at the dog's request really. This is epecially true when coming from children who can act strange (to dogs) and not be able to notice the body language asking them to stop.
If all that seems like a lot for you then also consider finding a good local trainer who can support you and work through this with you. It can really help to have another pair of eyes looking at dog body language while you interact.