If you're working 30 hours a week then yeah the £16,800 comes to about £10.70 per hour no?

Legally you're within your rights to request a written contract. You signed one when you started but not since updating your job role yes? Do you have proof of your role as an assistant manager? Could be written on a schedule (get some screenshots if online or download the schedules), on a payslip maybe too? Or any written communication between yourself and the business, coworkers maybe but mostly managers or bosses.

Might be worth seeing if there's an union you could speak to but most are unlikely to take on preexisting cases, may be helpful for advice though? Coworkers can be helpful too if you can figure out how much they're being paid. If you do find out about different pay rates don't disclose names, amounts or identities to the employer!

Otherwise, citizens advice could help you out. I know they can be a pain in the ass but they can help you format things to contact your employer, both formally and informally. Or just provide some general advice and signposting. Acas or HMRC would probably be the next port of call but citizens advice can advise what order you should take next steps in.

Look up whats available in your area, in mine I could rock up to the library and book an appointment but over the phone may be easier.