I think your situation is something that most of use have had to deal with at some point.
You say you can't afford to loose these clients, but can you really afford to keep them? They are costing you money, plus knocking your confidence.
I work from home, but I do have rules that I try to abide by. Firstly, I do not allow small children and use Health & Safety as my reason. Secondly, all my clients know that if they are more than 20 mins late and I have a full diary, then I may have to re-schedule their appointment.
It's not fair that some people treat us as if this is just a "hobby" and do not appreciate that it is our living.
The most cheeky thing is telling the children to sit in your lounge. This is your personal space and I would hate to have small children in my lounge if I am worning in another room. What if something of value is broken. Would the mother replace it?. And as for asking what food you have for the children, well I am gobsmacked at her cheek.
Don't let these people knock your confidence. Next time she phones for an appointment, just say that you have reviewed your policies and in future will not be offering any concessions or discounts. If she decided not to come back, then OK, you do not have the problem of being stressed out when you know she is due or feel resentful of the fact that her appointment could be replaced with a full paying client.
There are lots of lovely clients out there, but just a small few that spoil it for us.
Good luck & keep smiling
Mollylolly