I have an "integrated masters" degree. How can I eventually gain a PhD?
In the UK, you will be expected to have some research experience before applying to a PhD programme in science: not necessarily publications (although that always helps!), but certainly having done an extended project and written a dissertation about it. In my field (physics) you can definitely apply straight out of an integrated masters degree, provided that your last year had a substantial research component. 

If you don't have any research background at all (or just short summer projects or the equivalent), then this is the "extra work" you are looking for. Find a MSc by research in a field as close as possible to the one you want to do a PhD in. This also has the advantage that you get to decide whether a PhD is really for you while only taking a year, and to leave with a qualification at the end if it's not.

Of course, this is just general advice: as JeffE says, individual universities will be able to provide specific information on their admissions and funding criteria.