@Natasha45 As a host, you of course have the right to turn away guests who show up with more people than booked for, or otherwise have neglected to read that you don't allow children (or pets, as the case may be).
However, since you asked what others would have done in this case, I would have been inclined not to turn them away, seeing as how they didn't get angry or otherwise aggressive, nor were they trying to sneak another adult in, it was their child. Instead I'd have outlined the safety concerns regarding children, insurance issues, etc. saying that is why you don't accept kids. But that since they were already there and had neglected to read that listing info (you really don't have proof that she was lying about that, altho she may well have been), if they will assure you that the child will be vigilantly watched at all times, and that they make sure to leave the place really clean, with no crayon scribbles on the wall, cookie crumbs between the sofa cushions, or a urine.soaked mattress, and are willing to pay for the extra guest not mentioned (assuming you charge more for 3 than for 2), that you'll make an exception for them.
Of course, if I subsequently saw the child playing on the lakeshore without a parent next to them, I'd ask them to leave, as they were not doing as promised. And I would have mentioned in my review that they showed up with a child when your listing clearly states that you don't accept children, in order to warn other hosts in case it was a stunt they might try again.