The answer to this is yes, however you need to consider if this is the right move for you.

If you are already registered for self assessment with HMRC, you don’t need to register again.  You simply complete two or more self employment pages on the self assessment form (one for each business).

This can get a bit messy and you might just want to consider running it as one business that does two different things.  For instance, it can be hard to separate which business is paying for what if they’re operated from the same premises.

It gets a bit more complicated when it comes to the question of VAT registration.  If your businesses are connected and the joint turnover exceeds the VAT registration threshold then you will need to register all of the businesses for VAT.  In the majority of cases, HMRC will consider businesses to be connected if they are both owned by the same person even if they do completely different things.  It is only in exceptional circumstances that HMRC will consider them to be separate businesses and exempt them from VAT registration (if their separate incomes are below the VAT registration threshold).