Do I need to register as self-employed?

Yes, as a tutor you will need to register as a self-employed individual and declare your earnings. You can do this here: register as self-employed

The HMRC website provides guidance for registration, including contact details via phone and webchat. 

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NB: Please make sure you are eligible to work in the UK on a self-employed basis before applying. This may not be the case if you are from outside the EU or on a tier-4 visa and have not completed any tertiary education in the past.