+1 to what Amy says, but also think about having more than one account. I have my long-standing personal current account with one bank, then opened another one for my civvy freelance payments, and a building society savings account for my tax/NI money. So, I have three options for paying cash into.
HMRC give you some leeway between earning as self-employed and registering with them, and the bank would have no idea about that anyway. You've still got plenty of time to submit a return showing your earnings for the whole of 2017. If in doubt, get an accountant.
I would go arrange an appointment to open a new account, and say you receive most of your income in cash - you can say you're anything from a stripper/lapdancer to a mobile hairdresser, nail technician or masseur. Even a gardener!