Making payment to HMRC

All other things being equal he should be able to go back to 2006. Write back and ask to be reassessed.
 
It could soon be a lot easier to make the payment and allocate it to the desired years. From the "Your National Insurance Record" section of GOV.UK:

Important
You can soon use your Personal Tax Account to check if you can make voluntary National Insurance contributions, and pay using the online service.
 

So they will notify you every year and remind you to pay? That’s fantastic?
Has anyone set up a direct debit so that they can avoid missing a date? How did you do that?
TIA…!
 
You can't set up a DD from an Irish bank a/c.

You'll get an annual reminder & fire off the payment as soon as you get it.
Looks like a £Sterling DD can be set up from a multicurrency account, such as Wise. (I'm not at the payment stage yet, so haven't actually tried it)