Hi,
I hope you can get it all to work like you want it to, and then have lots more fun racing them with your son.
My suggestions are just top-of-the-head, rule-of-thumb style thoughts based on the little experience I have of little motors, I really couldn't know if they will solve the problem, please be aware of that. The 10R might need to be smaller to accommodate 0.8A start-up current, 4.8V/0.8A = 6R 1.2W, so 2W. But only a spike so, 4.8V/10R = 0.48A 2.3W, so 3W. 4.8V/0.3A = 16R 1.44W so 2W. As 0.8A is brief, a 2W is safe margin, and even a 1W resistor for all above currents should cope if motor on only in brief spurts, if on for e.g. >0.1s, use 2W or 3W, whatever the maths tells you, W = (A x A) x R.
Maybe before buying components that may not solve problem, if you dare, or know how to do it, taking a car apart and taking a photo of the internals and any pcb it has would/could be useful info. Bitter and/or highly stressful experience tells me taking something apart can be far easier than the sometimes nightmare of trying to put finickety devices back together. Even a computer mouse can be a small challenge, so a toy car may be harder to fit together again, or not - it might be easier, who knows.
Good luck!