You need to consider various things. First is use of your home. If a room is used solely for business then you are liable for capital gain when you sell. You don't need any checks on your home as a tutor but you do need insurance - or should have- for public and professional liability in case anyone hurts themselves in your home or tries to sue you for something. Presumably you are a qualified teacher and can provide references?
It's up to parents to check your home and make sure they are happy with a child being there.
And you need to register with HMRC as self employed and declare your income regardless of whether it's cash in hand.
Of course you have to pay tax if your business profits are above your personal allowance. You offset expenses (materials etc) against your income to calculate your profits. It doesn't matter if you are paid cash or paid into a bank account, you are still earning an income.