Up your budget and get a DSLR
They arent that more expensive and the sheer absence of shutter lag and use of higher ISO for lower lighting and indoor shots will ensure you get many more usable shot of the children. Children do not hag around to be photgraphed and most timesiit is indoors
Although there are some child specific techniques , getting the kit right is the very first step
Forget the "complicated" settings for now, DSLRs can be used as Point and shoot in Auto mode for starters
You might surprise yourself and get to know the more creative modes later
For little more than £200 you can get a second hand Canon orr eve a new Nikon D40
The only downside is size , you would need large pockets ..

Avoiding this route will only get you seeking for this sort of solution.. eventually,
I would give it a serious thought
Good luck

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