If the demand for payment isn't accompanied by a registered copyright, then you have no way of knowing if their client actually owns the image, and they've admitted their client has no copyright registration. Plus, the fact that they've admitted there's no registration is very important. They can't start an infringement lawsuit unless and until they get one - they'd have to expedite registration for litigation, which costs them more, and they hadn't bothered to do a normal registration when the work was created or before sending you this demand letter. Since they haven't regsitered this image, and your alleged infringement has already happened, they won't be able to claim statutory damages or attorney's fees, and they're stuck with actual damages, which are unlikely to be £983.25.
Your being sorry, it being an innocent mistake, and your lack of funds doesn't matter and probably won't matter to them, and none of that affects how much you've hurt the owner, if their client is the actual owner.
But yes, your bargaining position is much better than if they had proof of ownership and registration of the image, so don't be in a hurry to pay them the amount they demand, you should pay only a nominal fee based on the actual license fee for this image based on past use, if any. If they're in the U.K. and you're in the U.S., it's even less likely they'll pursue this claim.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
I agree with Ms Koslyn. I only want to add that, as a practical matter, you may want to ignore them until and unless they file suit. As pointed out, filing suit may cost them more than they claim to have been damaged.