No contract? Dodgy pay? Just go.
You'll struggle to actually get a proper reference out of anyone these days, particularly an employer, (as apposed to a mentor / manager / colleague) as the future employer can, in theory, use that against your previous if you don't perform.
slightly o/t: My o/h works in payroll for the Identical Blue Men. They pay over a month in advance and so have an odd problem that people, when they leave, have a habit of just walking out of the door, not saying a word and aren't heard from again. Their HR team supposedly have procedures to deal with this where they set up investigations, try and reach out to the leaver, do all the duty of care stuff you're meant to. Assuming the leaver has no contact with HR throughout this process they continue to receive full pay. This can run for three months or so until they finally run their termination routines, at which point they simply close the case. Tiz bonkers.