Some great information in the posts above about being easy to demonstrate you are not an employee and this is just a debt that has fallen due. Personally I would be inclined to write to them (agency)with recorded deliveries advising them they are overdue and have x days to pay in accordance with the terms of the contracts dated xx/yy/zz, if payment s is not made by certain date you will commence legal proceedings against the agency for unpaid debts.
You will probably get some BS response, but at least you have given them a chance, after BS response a second letter advising this is their final warning and they now have 7 days to pay otherwise the following claim will be filed With the courts. The. You could send a copy of the claim that will be filed with the money claim service, they will know you're serious then.
If you didn't have your money in that date then submit the claim online site mentioned above). I would suggest you make sure the claim is below the small claims court threshold of £10,000 GBP (https://www.citizensadvice.org.uk/law-and-rights/legal-system/taking-legal-action/small-claims/) so your claim can be assigned to the small claims track, what this means if you lose you're not liable for their legal costs (in most cases). Otherwise any other track and you would be liable for costs if you lose. If you have over £10,000 due e.g. Maybe 2 or 3 unpaid invoices I would be inclined to pursue one at a time to keep below the £10,000 limit.
Small claims court is relatively simple and you can represent yourself, you sit in a room with the judge and the party you're pursing and it's very relaxed, no court room dramas.
I'd simply draft a statement of facts, you worked through the agency from xx/yy/zz as demonstrated in your contract (exhibit/appendix 1).
The contract (exhibit 1/appendix1) clearly states the terms of the services you will supply for payment.
Exhibit/appendix 2is a copy of your work roster and confirmation of the completed work.
Exhibit/appendix 3 is a copy of previous periods of work you completed and your invoices were adequately paid (shows this is how it works and they paid historically) - don't need this but helps paint a clear picture.
In general I'd draw fact to the point that exhibit/appendix 1 makes it very clear what the terms of payment are, that you're not an employee ( as there is probably a term that says at no point are you an employee) etc
All of the above assumes there is no term in the contract for services that says the agency had to be paid for you to be paid ?
Your claim will also be for interest too on the amount claimed - http://www.compactlaw.co.uk/free-legal-information/small-claims-court/claiming-interest.html
Remember the small claims limit if adding interest, keep it below this.
If you can't keep below the limit, perhaps a group claim, if there are several of you in the same boat filing the claim again the agency not paying you, it will get allocated to a different court track, but you can all split the costs, if you win the other side side pays anyway.
I haven't got Legal training or qualification etc and is is just what I would do personally, I have represented myself in small claims historically and been through non small claims process too, I would choose small claims every time just to keep the process quick and cheap and easy.
If you win you can then demand payment, if they won't pay you take notice of your win to a judge by filling in a form and asking for a variety of enforcement methods, my chosen one was asking the judge to freeze the accounts of the business I was suing - you simply get the bank details the judge signs the order and it is served on the bank, eh voila the business can't do anything. There are a number of ways to get a firms bank details for the paperwork. This wil probably ruin their day and result in payment very quickly (list all their accounts on the order form).
Another thing I have learnt is when suing people someone may be thinking of folding the company and starting as someone else to avoid payment even if you win. A way to avoid this is to ask for the amout disputed to be paid into court, this means the defendant (if the judge agrees) has to pay the amount claimed or part of, into court so if they do decide to go bankrupt, liquidate etc, your money is sat pending payment to you if you win.
Good luck and I hope you get what is rightfully yours. It's been a few years since I was at small claims (2 years) I'm sure the process hasnt changed that much, it's good fun too and a lovely feeling when you win
Maybe a free half hour with a solicitor would be worth it, but also your home legal assistance cover might cover it too if you have the add on as part of home insurance (depending on the policy/exclusions).
p.s. Enclose copies of all previous comms with the company in your court paperwork/exhibits/appendices to show you have tried all things reasonable to get your money, also add in your comms to the agency you will only deal with them in writing from now on.
Pps - sorry if all the above sounds like teaching suck eggs etc.