Thanks for this additional information, that is helpful.
It is fairly typical for a vet to provide medicine such as steroids and antibiotics. This will help short term. However, once symptoms continue to persist, it is time to take a closer look at things.
The issue with over grooming is that there are many possible causes.
We won't be able to properly control symptoms until we know why she is doing this.
If she does have e.g. allergies, we are unlikely to be able to cure her and her signs will always wax and wane but we should get better control of things and be able to put a treatment plan in place. The aim is to have her as comfortable as possible.
To reach a diagnosis, it usually means carrying out a few straightforward tests.
This would generally be a skin scrape, skin swab and perhaps some blood tests.
It may also mean some trials such as an elimination diet for 8 weeks and avoidance of certain allergens.
For all cats who overgroom, we always recommend meticulous parasite prevention (a prescription product used monthly on all pets and routine house cleaning). It is important there is never a gap in this.
We should also start feeding skin supplements containing omega 3 oils and collagen
As her skin is weeping, she sounds to have an infection and will need antibiotics and likely also some more anti itch medicine to break her itch scratch cycle and allow her symptoms to settle.
For those with behavioural overgrooming, it can help to try to minimise stress (keeping a consistent routine, having hideaway safe spaces, using a Feliway plug in, starting her on prescription anxiolytics etc).
Her condition is very common and all GP cats should be well versed in caring for cats who overgroom; indeed, it is one of the most common reasons a cat is present to a vet.
I personally work in the Greater London area so do not have a recommendation for a Southampton clinic. You could ask for referral to a dermatologist or you could seek a 2nd opinion from a local clinic with good online reviews.