Letter to my MP - home office powers in the context of the EU referendum

In the run up to the EU referendum it is important that we operate as far as possible on the basis of factual information, which is not always easy.

I note from Baroness Neville-Jones's piece in the Guardian last Friday that about 6000 non-British EEA nationals have been turned away from our borders on security grounds since 2010.

On the other side, the Vote Leave campaign tells us today that 50 people with previous convictions have committed serious crimes while in the UK.  I assume that both these claims are fundamentally true.

In his statement following the European Council on 19 February the prime minister told us that he (and Theresa May I think) had negotiated:

1. new powers to stop criminals from other countries entering the UK, and to deport them if they are already here
2. longer re-entry bans for fraudsters and people who collude in sham marriages
3. powers to prevent EU nationals avoiding British immigration rules when bringing their families from outside the EU.

I have found it difficult to find any further information on these new powers, and would be grateful if you could help me fill in the gaps in my knowledge.