The contract
That's really not good enough for them to say they issued it. Virtually every self-employed contract for services is issued by one party and the other party signs and returns a copy. Its their word against yours.
In any event, what does it say about payment for services provided? How are the 'services' paid for - by time expended on outputs delivered or something else
The self-employed bit
Again, not very clever of them. I'll find and dig out the link to HMRC's indicators of whether a person is NORMALLY regarded as employed or self-employed - see how boxes your contract ticks. In the event that HMRC was to deem the employment as 'employed' status, the onus is then on the employer to pay not only the employee's and employer's NI but also the deemed PAYE tax - based on the amount paid to the individual. It is then an issue between the employer and the individual they attempted to employ on an SE contract to resolve the overpayment.
The terminology used by the company on your 'payslips' (which they are not of course, they are remittance advice statements - but you didn't issue an invoice?) does tend to suggest they were attempting to treat you as a SE contractor - but they have to satisfy a lot more than that.
I'm not sure it is worth irritating them by counter-claiming that you will invoice for your expenses - these should have been negotiated as part of the payments for services rendered.
I wouldn't by inclined to put any of my points in writing to them. Just be prepared to go along to a one-to-one discussion (of which you should write your own notes. You could then put some of the 'muddling' consequencesa of their actions that I have suggested to you. You could present them not as facts, but as 'it is my understanding that'. The objective of the meeting should be to part at the end with both sides agreeing to take things no further.
http://www.hmrc.gov.u...ment-status/index.htm