Can you apply for a passport with your deed poll documents?

Yes - the Identity & Passport Service will accept a deed poll as proof of your change of name.

However a deed poll is not a proof of identity on its own. You'll still need to send in your birth certificate (or your previous passport, if you've got one).

If you've changed your name by deed poll, your new passport won't mention your old name - it'll only mention your new name (as stated on your deed poll).