clear and obvious human error that would be reasonable for anyone to make,
It's things like this that they dine out very generously on !
Clearly it's your error not theirs, so any reps you submit need to be non-confrontational and reasonably humble, although not "crawling on your belly" type. You will be asking them to use their discretion in view of this simple human error. It is entirely up to them whether to accept your reps or not. They have not lost any money at all so morally, should do so. Be aware that adjudicators can only decide on a law basis, and cannot consider mitigation. This arose from a High Court case a long time ago, probably in the 90s when decriminalised parking enforcement came in.
You could try something like this -
Dear Sirs
Re PCN <number> dated <dd/mm/yyyy> served at Ducie Street
I wish to politely request that the above PCN be cancelled as it only arose as the result of a very minor error when entering my car registration letter into <name of pay-by-phone app>
On the date in question, I parked on Ducie Street to take my girlfriend out in Manchester city on the ocasion of her birthday. I paid to park using <PbP app> for a full two hours and was therefore very surprised to find a PCN on my car when we returned from lunch within the two hour period. A quick examination of my PbP entry showed that I had made a very minor typing error when entering the registration number, typing XX
61 XXX instead of XX
71 xxx. An easy mistake to make, and I hope you will agree that an error like this, particularly when the council have lost no money at all from my error, is minor and does not require the imposition of penalty. I hope therefore that you can cancel the PCN as a goodwill gesture. I shall make sure I am more careful in future.
Yours faithfully