Not read all the replies, just @HenriettaHippopotamus but actually you do have a contract - it has been created by mutual conduct over a 3 year business relationship in which you have provided services in exchange for payment. It was regular - 4 hours a week.
A contract does not need to be a written document under UK law: it comprises of three main components: offer, acceptance and consideration. You offered your services, they were accepted and consideration in the form of payment of your fee has been forthcoming. This persisted for 3 years so you have implied contractual terms. Whilst you have no formal defined contractual agreement, stating cancellation policy for example, ‘reasonable notice’ would normally be accepted as an implied term by a court of law.
I appreciate that the stress of a county court judgement may be something you wish to avoid, but you can get googleAI/chat GTP to help you draft a ‘letter before action’ - stating that a contract for services has existed between you for 3 years, evidenced by their receiving your services in exchange for payment, and that reasonable notice ought to have been given. You would therefore request that 4 weeks income (4x hours) be paid for by X date after which you will have to take the matter to the small claims court.
You don’t actually have to proceed, but they may be guilted into understanding what they have done and maybe at least come back and offer you half if not all of it.
And re people saying ‘word gets around’, ignore that. You have a right to expect to be treated fairly. The chances of this family impacting whether you get more clients or not is minimal. The home ed community is small, the number of experienced tutors is even smaller. No one I know goes on recommendation of friends for tutors (I have formerly homeschooled) - they research on tutor directories, research the tutors to check credentials and proceed from there. And if anyone ever asks about them specifically, because they have said something, just be honest: you tutored their child for 3 years and they cancelled without notice and you had not choice but to insist upon payment. As a result you now have a strict written contract in place so that everyone is very clear of expectations. The family won’t say anything, though, because no-one likes to be judged and risk censure from their friends, so the chances are they will keep it to themselves and potentially just pay up.
As they say - if you don’t ask, you don’t get. So, yes, you state that they owe you for cancelled tutor sessions and lack of reasonable notice.