Unsure whether to start new job (sort of!)

So I work in education via an agency. A few months ago, around March time, I was successful at interview for a job in the school I work in via the agency. I started 2 weeks later but haven't actually been "put on the books" in the school. It was one thing after another in terms of sorting out the contract. I heard nothing for 4 weeks, then it started. They wanted so many different things, even things that the job advert didn't require/ask for (would also like to point out that the other people in my team did not need to provide these when they started). But they didn't let me know the things I needed and I've had to constantly keep asking every day what else they need. Was told I'd be able to start after a DBS check. That came and still they kept adding the things they needed before I could start.

I will say that yes, I get paid via the agency. But I haven't been paid over the summer. I started the job in March and have been doing it every single day. I waited 5 weeks for a reference from the person who gave me the job. I just feel so messed around. The team I work with, they all got paid over the summer. I'm doing the job and I haven't been paid. I now have to wait until September to see if they have everything they need so they can set a start date. But it's literally the case that I'll go from July 27th to September 28th without pay. For a job I'm doing like everyone else.

I'm starting to question whether or not I should even go back/start this new job. Considering I didn't actually want the job and they only asked me after 2 other people turned it down. Tbh, I also find it boring and I'm afraid that when I go to apply in October for a PGCE, I'll end up not being able to because I won't be able to get a reference.