Part objective opinion here, part devil's advocate; but I think you're mixing a few different issues here.
First, I don't think you should assume that they don't understand the importance of their children's education or that they don't care about it. They're already paying for a private tutor, which at least says a little something about them. If the husband doesn't allow the wife to drive (or whatever he doesn't "allow") you can't assume that the same husband doesn't tell her to get the shopping done when he's not around. You can't assume that because the woman doesn't work or doesn't have a degree, or that because English is her second language, that she doesn't value, or know, some things. You can't assume she isn't a mother who doesn't want to leave a child alone, or her house, with an "outsider" (we've all seen those cases on the news; this lady may not leave her son for reasons very different from what you think). That's not even saying she doesn't trust you. It's saying that her head tells her that even if she does she shouldn't take any chances with her child.
Basically, you're doing what a lot of people do, which is to extend beyond the problem at hand and on to imagining what's in someone else's head (or not in her head). She could be someone who figures, "I'm paying this individual to do work for me. I have a right to cancel." She may or may not know any number of things. That doesn't matter and isn't really the point. Some people have that "attitude" when they're paying for a service. She may be thinking, "Teach my kids when you're here. Don't worry about anything we do or think beyond trying to teach the children the subject."
I think you have to look only at your own problem and her behavior with regard to cancelling. No matter how much you need the money, you could (if you were willing to do without the money and can't find income from anywhere else) quit. Whether you can find other income really isn't her problem. So I think you have to decide to either keep the income from her or do without it.
If you decide you want/need to keep it you have three options: Telling her you'll be charging for cancelled appointments (and risking her telling you that she'll find another tutor), asking her to give you 24 or 48 hours notice "so you can rearrange your schedule" (and either charging her for cancellations too close to the appointment time, and risking her refusal to comply or else her thinking you've "too picky" and dumping you), or unhappily accepting that she does what she does, teaching the kids when you're there, not spending your own brain-time thinking beyond just trying to teach the kids when you're there, and keeping in mind that a lot of people who need money badly take "lousy gigs" in order to earn it.
If she's waiting until you show up at the door, or until an hour before you're supposed to be there, to cancel she's being really thoughtless and disrespectful. There's no doubt about that. I'd think you could politely ask her not to wait so long before cancelling (if you haven't already done that). I'd think she can understand that much. If that works, great. If it doesn't (and she either refuses to comply with the request or else tells you she'll find someone else) you have to either decide to accept the lousy treatment with the money, or else let her find someone else.
I think, when it comes down to it, the only thing you need her to understand is that driving so far for nothing is a problem for you. You can't know what other factors exist in her life that may actually make that food shopping more of an emergency that it would appear to an outsider. You can't assume she doesn't understand what missing a lesson might (or might not) do to her child's education; because a whole family's worth of kids who need tutoring indicates that there's stuff going on besides one kid's missing a few math lessons. There's a chance these kids aren't going to benefit all that much from tutoring (I hate to say it, but kids with too many school problems often don't benefit a whole lot even one-on-one attention). This lady may know her kids aren't getting much from the tutoring, so she maybe that's why she a) feels so free to cancel, and b) may not respect you as much as you really deserve to be respected.
People like social workers and teachers who deal with some kinds of families often burn out because they discover how futile it can be. I think the only problem you should be looking at, addressing, and/or unhappily living with is the simple matter of cancellations and inconvenience to you (and the not-so-minor matter of not being treated with respect).
With all the other "surrounding" stuff, I think you're likely either under-estimating the woman out of not understanding what she lives with, worries about, or sees for herself; or else over-estimating your own potential impact on the overall education of those children.
These people are handing over money that they either can easily afford or that is difficult for them to afford. If they can easily afford it it's probably safe to assume they're not completely ignorant and stupid people. The fact that the wife doesn't work doesn't mean she's stupid, lazy, or ignorant. On the other hand, if they can barely afford tutoring (which actually could be the reason for cancelling so often), they're apparently trying to keep the kids from failing and/or being in the Special Needs program. It looks as if they already do care about their children's education a lot more than you may think. Then again, if the only reason they've gone the tutor route is that they wanted to refuse Special Needs help without having Children's Services brought into the mix; well, maybe, if the kids get nothing from tutoring the next thing will be that social workers get introduced to the picture. (Maybe that's what they need if they're as clueless as you seem to think they are.)
Either way, when people are in "desperate" need of money they run that much more risk of being vulnerable to burnout; so between your profession and your apparent financial need, I just think you need to let a bunch of the "extra" stuff go (if for no reason other than aiming to lower the risk of emotional burnout). Living with cruddy situations isn't easy, but stewing over them can make them worse.
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