I am not aware of the parking regulations for this car park, but, from your messages, I understand that you: a)you followed the parking regulations in order to confirm that you were there shopping and b)you did not exceed the allowed parking allowance, let alone leave the car parked overnight.
If it's so, and you can prove this - also by a statement with the card transactions confirming the date and time of the shopping and by a witness statement of someone who was with you - you may appeal the charge (the link should be the following, but please check: https://nexusplatform.co.uk/pcn).
If you have received the parking charge notice only on Saturday, you have another ground for appeal because, being this a charge for parking in a private area, the notice should be served on the registered keeper within 14 days of the day after the parking ended.
Some delay could be allowed if the DVLA records are not updated, but it seems that it took too long to serve this notice.
Based on the following advice that I can find in the FAQ section of their website, I recommend that an appeal is submitted within 14 days of the date the letter was issued:
What will happen if I ignore the Parking Charge Notice?
Failure to pay or appeal the PCN within 14 days of the date it was issued will result in the early payment discount no longer being applicable to your case. The amount payable will increase to the full amount due, which is shown clearly on the PCN and is also displayed on the signs.
If no payment or appeal is received within 28 days of the date your PCN was issued, the case may be escalated for further action. This includes, but is not limited to, Debt Collection and may ultimately result in Court action and further costs may be incurred.