Why can't I get a job with sponsorship in the UK?

I'm a business grad with over 10 years of experience in the Middle East & the UK. I do currently have a job as an Operations Manager but i feel I'm underpaid and in a sector where I've already hit the ceiling and not the best for career progression.

I've been applying for jobs for over 5 months and the job search just doesnt seem to be going anywhere. I've been sent rejections with the sole reason being we can't offer you a sponsorship and this is consistent with both public and private organisations. Also, I can't seem to get any responses pushing past the £35k-£40k mark which I guess is more of a fresh grad salary in fields such as project management, operations, facilities management, property management, change management, digital transformation, agile delivery (CSM) and solutions based roles. Can people recommend any employment agencies say in cities such as Manchester, London, Bristol, Leeds; if this is allowed?

Any insight on this matter is appreciated.

Thank you.

TLDR: Business grad with 10+ years' experience, seeking roles above £40k. Struggling with sponsorship barriers and limited responses. Any agency recommendations in major UK cities would be appreciated;if thats allowed.