Flatbread's point is important. Grown ups often "test the waters" by saying provocative things to see how they will play. If they get a positive or no response, they'll take that as an endorsement of their views. Children and young people do the same thing.
If you're in some position of authority with a young person (e.g. tutor, scout leader, youth worker, auntie, etc.) then you have the opportunity to engage with them, find out where the view's coming from, present alternatives. If they are parroting something they've heard from a parent or friend, they might just be saying it because they think they "should," and may not even understand what they are saying.
No one should ever, ever put up with another person's bigotry just because they are being paid to provide a service to them. I'm 
at those who think that this is acceptable. And no, it doesn't have to be as direct as saying "I hate Jews," to be bad enough to do something about it. I also think OP you should feel the need to "come out" as a Jew, any more than you should feel you have to "come out" as a Lesbian in response to a homophobic remark, or declare a disability if someone makes a demeaning comment about disabled people. If you felt confident doing that, that's okay, but it sounds like this kind of hit you like a slap. Saying you're Jewish might give her pause for thought if she likes you, but it might also give her an "excuse" for not following your direction in future or a kind of "ammunition" to hold against you if something doesn't go well in the tutoring relationship.
Perhaps if a similar comment or response comes up in future, just ask what she means by it or why she thinks such and such. Don't be afraid to say that you find her comments/views hurtful, perhaps saying that you have a dear friend with a name like so and so or a neighbour was attacked by a person who said they didn't like such and such. It doesn't have to be heavy handed or preachy, can still be on a personal level but not leaving you feeling vulnerable.
If the situation continues to grate, and you feel it's worth the risk of losing the client, it could be worth talking to the parent(s) and explaining that you feel uncomfortable/unhappy in the situation. It could be they are unaware she is saying things like this (and might WANT to know - maybe they suspect a vengeful ex or controlling older boyfriend is planting ideas, I dunno . . . ) Of course it could be that they are card carrying members of the British Nazi Party and will sack you on the spot. If so, you'd probably not want to stay on anyhow!
Ah, and one other thing. You mentioned that you were surprised the girl held these views because she is from a mixed cultural and possibly mixed ethnic background. Unfortunately, in my experience, that isn't necessarily a guarantee that a person will be more accepting or understanding of difference in others. If they feel conflicted about their own identity, they may actually seek to deny their difference, and one way of doing that is being very negative about other folks who are "different," as well.
Best of luck whatever happens.