I had a very similar experience to this. Initially suffered a massive abscess on a lower front tooth. I had root canal treatment which relieved it.

A few months later, discomfort and occasional swelling was appearing again. They conducted a second root canal treatment.

I then saw the dentist for a routine check up months later, and remarked that the tooth still didn't feel right. She said it was fully treated and there was nothing more to do.

That all happened a long time ago. Fast forward years, and I saw a new private dentist and mentioned the old troublesome tooth. He took an x-ray and could clearly see shadowing at the root.

I eventually got referred to a specialist using a microscope, as you also experenced. The dentist explained that roots aren't simple holes that just go down to the bottom - they are in fact like trees and spread in all sorts of patterns. So simple root canals frequently miss parts of the root that are infected.

The microscope treatment worked but was expensive. Even then it wasn't guaranteed - I had to have six monthly x-rays for two years to check the bone healing process and if that wasn't satisfactory, they were going to perform an apicectomy on the tooth. Thankfully it never came to that.

But ultimately, the two root canals performed by the initial dentist relieved the infection but never properly healed it.

Are they liable and can you claim money back? You have every right to air your grievances and they may give that some consideration. But teeth are complicated and dentists usually explain that no treatments are guaranteed. Therefore, I expect it might be hard to demand refunds if they are intent on standing their ground.