PhD studentships from the North West Doctoral Training Centre
ESRC-funded studentships are available as part of an annual competition, funding social science projects that can include psychology, mental health and related neuroscience. These awards require the student to identifying supervisor(s) and submit a PhD project proposal. If you don't already have a Masters degree, you would have to apply for the 1+3 award which involves enrolling for a Masters as the first part of the studentship.
Another NWDTC pathway is CASE studentships (under "collaboration" on the same website). These are submitted by the prospective student's supervisors but can name a student on them. These require an additional non-HEI industrial partner such as a private company, with which the student spend at least 3 months per year working, and the rest in University.
Suitable research topics might include (this is not an exhaustive list):
The Institute of Psychology Health and Society at the University of Liverpool has an active Research School offering postgraduate study leading to the degree of PhD or MPhil.
There are a number of specialised PhD training programmes available within the Faculty of Biology, Medicine and Health.