Contract law is quite tricky in the UK. There doesn't need to be a piece of paper - such as a purchase order, agreeing the terms of the sale. A simple transaction in a shop for example, of a seller offering something at a price that a buyer pays for, and crucially - the seller accepts that amount *before* realising an error, is enough to deem that transaction completed because the seller offered at a price that the buyer paid. The seller has no recourse should they realise afterwards that they made a mistake with the selling price.
If there is a formal document in place agreeing the selling/purchase price, then that would take precedent over an invoicing error as an invoice does not supersede an agreement to purchase at a stated price.
They grey area here is that the buyer queried the amount with the seller before paying and the seller agreed the (incorrect) amount with the buyer, who has then paid that incorrect amount. Transaction complete.
I can see both sides of this, but usually when there's an invoicing error there isn't an issue between buyer/supplier - it just gets corrected.
Yet, it would seem that in this case, you should take formal legal advice if you want pursue the shortfall.