u/Akirax69

Hello there!
I am an Italian psychologist (master degree, post graduate courses, and currently holding a full membership at the BPS) planning to move to the UK.
At the moment I have no right to work in the UK and as far as I understood the best way to get in is to find a sponsorship VISA. In order to do so I am looking for jobs on the NHS as assistant psychologist but so far I only get rejected even before an interview. Most feedback point out that employers are filling vacancy locally before looking for an oversea candidate.
Therefore I am kinda stuck in this loop of sending out CV and getting rejected.
Do you have any consideration or suggestion to improve my chances to get a sponsorship?
Did anybody have a similar experience and found a way in?
But mostly am I doing this wrong?