I kinda like what I've seen of the new Pentax 35 for this, it's not truly wide, but that can be a bit tricky for portraits, anyway.
Chasing young kids around in informal settings is the kind of thing I actually like a zoom for, though, if there's enough light.
Kind of the best kid shots, I think, you get when you kind of go a little ways away from the parents, (Or in your case, other parent or any point of interest) and just let the kid run around: they'll tend to run between the folks, funny lady with the camera, and whatever else catches their attention out there, and you get shots aplenty, especially in a pretty setting. They'll pretty invariably settle down/get a little bored and then you might get some sort of calmer shots.
Zooms are good for this, cause you can frame up pretty quickly, and playful works. If some of the skills aren't second nature, you may miss a bunch, but you'll probably get something good with all this automation. Just keep shooting.
(And if you're outside, there's really nothing wrong with that kit lens for the job. A little longer might be nice, but it's more than wide enough.
And come to think of it, the quick-shift focusing could be a real asset there.
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