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In which case I'd reorder and rephrase your informal reps.

Surely you're trying to set out a compelling argument for the authority to exercise discretion(could be 'circumstances beyond your control...but I'm not going to get too exercised about the distinction), therefore perhaps....

On *** my son, aged **, developed a temperature and therefore I went to the (surely local, although you've said that you're not a resident of Richmond) pharmacy to buy some ***** (IMO, medicine is too vague). I parked as can be seen and saw the CEO to whom I explained the situation and who KINDLY! allowed me to park for 5 minutes without paying.

As luck would have it, **** pharmacy could not accept card payments and therefore as I did not have cash on me I had to go to the nearest ATM and then return to the pharmacy. This detour caused me to be late getting back to my car where I found the PCN, perhaps understandably.

Given the circumstances, I would ask the authority to exercise discretion on this occasion. I would also suggest that the actual wording of the contravention cannot be correct in that the parking place accepts phone payment and therefore not displaying something whose display is optional cannot be correct, surely. Likewise the sign doesn't reflect the option to pay by phone, but I hope that it won't be necessary to examine these issues in further detail in this case.

Some thoughts. Pl fill in the blanks and post a draft here first.

Two errors IMO:

The sign is incorrect;
The contravention is incorrect.

I would start with your account, the CEO's conditional permission and that for reasons beyond your control it took you longer to get back to the car than agreed with the CEO*. In this regard, you therefore parked without permission or having obtained parking rights. On which point, I would draw the authority's attention to the following:

The traffic sign in evidence states: Pay at machine. Display ticket.
The machine accepts payment and advises that parking rights may be obtained via RingGo, which is also confirmed on the council's website.
The alleged contravention is that of failing to display a ticket, NOT failure to pay.

 It therefore follows that:

The traffic sign in evidence is incorrect because display of a ticket is not an obligation; and
The contravention is wrong for the same reasons.

Either constitutes a statutory and therefore the PCN must be cancelled. If the council wants to enforce this bay then they must amend the traffic sign and use a mandated contravention description.

I am quite angry at the CEO as I feel he acted in a deceitful way and actually intended to issue a PCN all along.

Why? According to you he gave you conditional permission to park where you would otherwise not have been permitted unless you paid, but you did not comply with those conditions.

What has getting money from an ATM got to do with your son's condition or your parking, you weren't paying to park anyway(not that you'd have needed cash).

There are some free parking provisions for residents of Richmond, do you qualify?

https://www.richmond.gov.uk/services/parking/parking_machines/parking_machine_guide