My dd was in a similar situation with a cafe job. The boss tried to pay her less than mw, which she refused and reminded him of the actual rate. Which he proceeded to pay her.
Poster, if your DD wants to be on the payroll she needs to ensure that she is. Just because she's being paid min wage doesn't equal being on the payroll. If she is, it'll mean she's accruing holiday entitlement, which she'll have to be paid for if she leaves without having taken it, she'll be able to get SSP if she's ill and off work longer than 3 days, be entitled to maternity pay, employees rights surrounding redundancy, health and safety laws etc. She needs to understand whether she has all that or not and decide whether she's happy with it, not assume she has it just because she's paid min wage.
She has no contract either
Of course she doesn't OP, she doesn't officially work there! There's not going to be a contract, ever. Other than whatever verbal agreement they've come to, that can't be proved to exist and is worthless as a contract. I'm beginning to think you don't understand the situation at all.
These jobs are done on the basis of trust. Employee trusts they'll get paid the agreed amount and employer won't ask any awkward questions about their circumstances or grass them up to anyone. Employer trusts employees will show up as agreed and won't try to sue them for accidents at work or cause them any other kind of trouble. Sometimes it can work out fine on that score. Other times the employer can totally take the piss on all aspects, secure in their suspicion that the employee won't want to rock the boat or risk losing the job by arguing back because they're not in a position to just go get another job.
To whomever said she should get minimum wage at least, in lieu of employees rights - usually those working off the books are doing so because they're upto no good themselves, so it's a mutually beneficial arrangement to both employee and employer. Lots will either have no legal rights to work in the UK so can't get a PAYE job. Or else they're claiming low/no income benefits, so to have an undeclared income below NMW suits them and leaves them better off financially overall than if they got paid NMW but lost the benefits.
There are some industries where the employers prefer off the books employees so they can shirk their legal obligations, including paying a lower wage, and it can result in a situation where nobody can realistically get a legal job in that industry at the lower levels. Which is a situation that unfairly disadvantages honest decent people who aren't doing anything wrong and just want to earn a proper legal living so they can pay their bills. That's my main issue with the existence of these jobs.
Lots of people consider working off the books immoral because even if you're doing nothing wrong, the employer is, and you're part of keeping that system alive and condoning it. So it's not something you go shouting about. For one thing, all it would take is for someone to report your employer - then if they end up closing down, because their business wasn't actually viable if it had to pay proper wages and comply with all laws etc (and now the fines or other legal consequences for not having done so), then you're out of a job. All because you couldn't keep quiet.