Thanks.
Your initial appeal was made on the basis that Horizon had failed to give notice of keeper liability as prescribed by section 9(2)(f) of PoFA, but looking at their notice, they have done - the back page that you have now uploaded would seem to give notice in the prescribed way. As far as I can see, the notice is PoFA compliant, so they are able to recover any due charges from you as the registered keeper.
So, for you to challenge it, we will need to look at other angles. Assuming the driver was a genuine Tesco customer, and not simply trying to use their land for some free parking, you could speak to the management of the store (ideally providing a receipt from the driver's visit) and ask them to cancel the charge. If you can acquire photos of the signage, it would also be useful to see this, to see if it provides any useful avenues of challenge.