Tax confusion/locum agency PAYE

Awrah

Hi there. 

I'm a locum pharmacy technician. 
I've recently thrown my hat back in the pharmacy ring (February) so I haven't really worked so far in this tax year, mainly due to hating pharmacy!
This means I haven't touched my 12500tax free allowance.
I resigned  to a locum agency I had used a few years ago under their PAYE and I've been working to make sure I get my £12500 free tax. 

My issue is that I feel like they're messing me about with payment and the timings of payment. 
I'm being told by my agency that they're having trouble with NHS professionals in regards to the pay code I work under as my rate of pay they agreed to is high. This is due to my decade of experience in prison pharmacy and my familiarity with the trust as I worked for them for 6 years as a locum.
The agency agreed my rate and I actually landed the job in their laps as I was asked by the bosses in the prison to help them out. 

I'm getting no assurances from my agency about when I'll get paid, my agent just says we'll try to get it sorted for you which is useless info. It's hard to communicate with them also.

So what I would like to find out is... Can I claim back the loss in tax for time worked Feb to Apr 2021 if I end up getting paid after tax year end? 
Or can I claim against my agency if they don't get this sorted? 

So sorry for the wall of text. I'm really anxious about the situation.