What are my housing rights

Smileless is absolutely correct. You have been given emergency housing and have the hope of permanent housing before long. If your friend moves in you could be turned out of the emergency housing and put way down the list for a permanent place. Don’t risk worsening your position.

My advice is don’t! I would wait and think very carefully about cohabiting until you are in more secure housing situation yourself. Dd nearly lost her emergency flat due to similar circumstances.

Make sure that if he moves in you're not breaking your tenancy agreement as this could make it very difficult for you to secure alternative accommodation in the future.

Why can’t you move into his place

I would think you are very lucky to get this tenancy and should not do anything which may affect it in the future.
Can you not put up with having your relationship as it is now until you could apply for your partners name to go on the new tenancy when it is offered to you,you should not take the risk of being homeless again and be grateful for what you have at the moment, best of luck for the future.

I think you might find it very difficult to get anyones name added to your tenancy, you were given emergency accommodation because you were homeless, if you had told them you were going to live with someone I expect they would have told you to get a private rental.

If your lease states that it’s for you only then that’s your answer.

suggest you contact shelter for specialist advice.

england.shelter.org.uk/housing_advice