I have worked alongside Mothers who stay home & these are some of the roughest jobs for me.
A lot of it IS because the comings + goings of the Mother does change the dynamic throughout the day.
Children do in fact act different w/their caregivers when the Mother is around.
It is just not a good dynamic.
I would just let the Mother know kindly after your trial days are complete that you think you would be better matched w/a position where you are completely on your own.
After all, this WAS a trial and you are not obligated to stay on.
Plus it's not like you are leaving anyone High & dry w/no childcare and that they cannot go to work since this Mother doesn't seem to work.
Truly...
No harm, no foul.
Good luck!