Adding a middle name by deed poll is entirely possible. Whether you want to add a family name, honour someone, align your name with a cultural or religious tradition, or simply add a name you have always wished you had, a deed poll is the legal instrument that makes it official. The process is identical to any other name change: you declare your current full name and your new full name (including the new middle name), sign in front of two witnesses, and the document is valid.
Your deed poll should state your complete current name (first name, any existing middle names, and your surname) and your complete new name, with the new middle name added. This makes the document unambiguous for any organisation that reviews it. A deed poll that clearly states both names, before and after, will be accepted by every UK organisation that processes name changes. For removing or swapping an existing middle name instead, see changing your middle name.
One thing to be aware of is that different organisations handle middle names differently. The Passport Office records middle names in full on the biographical data page of your passport; they appear as part of your full name. DVLA driving licences do not typically display middle names. Banks vary considerably; some record a full name including middle names, others store only a first name and surname. HMRC records middle names in your full name against your National Insurance number.
This inconsistency means that adding a middle name may have a visible effect at some organisations but not others, depending on how their systems store names. For the Passport Office, it is most visible: your passport will show your full name including the new middle name. For the DVLA and many banks, the middle name addition may not appear in the visible records even after the update.
This variation is not a legal problem; your deed poll establishes your legal name regardless of how individual systems display it. Don't be surprised if, for example, your bank doesn't add the middle name to your account display.