That's an excellent draft, I'd just suggest that before the last paragraph you add:
As my vehicle broke down prior to the start of the restricted hours, I was prevented from moving my vehicle owing to circumstances beyond my control and I made all reasonable efforts to summon the assistance of my breakdown company. It follows that the contravention did not occur.
Remember to keep a screenshot of the confirmation page.
Well there's a lot going on here and you have to be careful not to say anything misleading, but you are not under any duty to incriminate yourself either.
Here's a simple representation:
Dear Manchester City Council,
At the time of the alleged contravention my vehicle had suffered from a breakdown owing to a fault with the cooling fan. I had to wait a number of hours for an RAC patrol to attend and the vehicle could therefore not be moved from where it was owing to reasons beyond my control.
The RAC did eventually attend and were able to get my vehicle going again, though they advised me to arrange for a permanent repair, I attach a copy of the RAC report for your records.
I trust that in light of the above you will agree to cancel the PCN.
Yours faithfully,
While we wait for a response to that I'll get hold of the traffic order for this road.