A deed poll can be used to change any part of your name — including your middle name. Whether you want to add a middle name you never had, remove one you have always disliked, or replace an existing middle name with something different, the process is exactly the same as changing a first or last name. You declare your current full name, declare the new full name you intend to use, sign in front of two witnesses, and the document is legally valid.
One thing to note is that middle names are treated inconsistently by different organisations. Some (including the Passport Office) record middle names in full; others (including some banks and the DVLA) do not record them at all or store only the initial. This means a middle name change may be invisible at some organisations — they simply have no field to update. For organisations that do record your full middle name, you present your deed poll exactly as you would for any other name change.
If you are only changing your middle name, your deed poll should still state your complete current name (including first and last) as the name you are abandoning, and your complete new name (with the new middle name) as the name you intend to use. This ensures the document is unambiguous and will be accepted by any organisation that reviews it.