(UK) Recently offered a new job, not entirely sure if asking for more pay is acceptable

Not entirely sure this is the right subreddit but hey ho.

I've recently gained employment as an assistant psychologist in a private healthcare company in Scotland. I currently live in the North of England so this requires me relocating.

The pay for this role is just under £19,000. However, if you are employed as an assistant psychologist in the NHS, you'd start at £19,409. I've worked an equivalent role (band 4) in the NHS for 1.5 years, so I used to earn £19,852 (+ weekend overtime, so I think I earned around £21k). If I joined the NHS now as an assistant psychologist, I'd earn £20,551.

So basically, in this new role I'd be earning £1k less than I did a year ago in the NHS, and since then I've gained a masters and more experience. I'd also be getting 7 days less annual leave.

Is it possible to ask for more money? As I've always worked in the NHS I've never been able to negotiate pay. How would I even go about it without sounding rude / uncouth? This role is super competitive and I'm scared they'll retract my offer if I ask for more money.