"Plan A" as it is often called, is to speak to the management of Tesco to see if they will intervene.
APCOA are generally fairly easy to beat, so you could appeal as the keeper, being very careful not to tick any boxes or select any options that reveal who was driving. If in doubt, choose 'other' as the appeal option, and add text along the lines of the below:
Dear Sirs,
I have received your Parking Charge Notice (Ref: ________) for vehicle registration mark ____ ___, in which you allege that the driver has incurred a parking charge.
I am appealing this notice on the grounds that firstly the contravention did not occur, and secondly that even if it did, which is denied, I have no liability for this matter as the registered keeper of the vehicle.
On the date in question, the vehicle made two separate visits to Truro Tesco, the first at around [TIME], and a second at around [TIME]. Both visits took no longer than [TIME] each. Your ANPR system has failed to properly record this and has incorrectly recorded these two visits as one continuous stay.
Furthermore, I note from your correspondence that you are not seeking to hold me liable as the registered keeper, under Schedule 4 of the Protection of Freedoms Act 2012 ("The Act"). You have chosen not to issue a Notice to Keeper in accordance with The Act, and it is now too late for you to do so.
There is no obligation for me to name the driver and I will not be doing so. As I have no liability for this charge for the reasons outlined above, I look forward to your confirmation that my appeal has been accepted and you will not be seeking to recover this charge from me. If you do not accept this appeal you must issue a POPLA code.
Yours
If appealing online keep an eye on your spam emails and chase them up if they do not respond.
Its also worth noting that they spelled my surname incorrectly.
They will have got your name and address from the DVLA - it could be an error at APCOA's end, but it would be worth getting out your car's V5C document and checking your name isn't spelt incorrectly on there. If it is you ought to correct this with DVLA.