She cannot be your tenant, because she does not and never has, paid rent.
For the same reason, she cannot be a lodger.
For the same reason, she cannot be a licencee.

So, the only basis on which she can occupy the property is as your guest.

The fact that you may have split the grocer bill 50/50 is probably not relevant, as you have no obligation to feed either her or her child.

I suggest you set a date on which you require her to leave, record that in writing to her and serve it on her. Ensure you have a witness, both to service and content.

It might be worthwhile employing a solicitor firm to do that.

After that date has passed, she is a trespasser.
Unfortunately, trespass is a civil matter, so if she does not go voluntarily, your only option is to apply to the court, for an Order compelling her to leave.