Questions regarding voluntary NICs for someone who has deferred their State Pension

I have an elderly relative who needs help with her pension application.  She has recently recovered from a prolonged mental health episode that has prevented her from dealing with most financial admin or asking for help with it until now.  It turns out that she has not applied for her State Pension despite having had no income and limited savings (but just enough not to be eligible for Pension Credit).  She is 71. Having checked her NI record, it emerges that she has only 20 years of contributions, but can still buy 12 more between now and 5th April.  We have requested a calculation from DWP but it will probably not arrive on time.  We will likely end up having to send a cheque, unless we can get hold of the 18-digit code from HMRC. We can see how much the voluntary NICs cost by checking her record via Government Gateway. She reached State Pension Age in July 2018 and can thus only backdate her pension 12 months.  However, if she does this, any lump sum she gets will only be 62.5% (i.e. 20/32) of what she would get if she had made the voluntary NICs before reaching State Pension Age. Both the decision to defer and incomplete NI record can be attributed to her poor mental health and it seems unfair that she should be penalised so much for either.  Her condition and her resulting lack of engagement with her financial responsibilities has already cost her dearly in myriad other ways.  She has also been unlucky with many factors outside of her control, e.g. raising of SPA for women from 60-65 and pension rule changes in 2016 which have limited how far she can backdate. My understanding of her State Pension Entitlement is not 100% certain, but I believe that she would be receiving £126.40/week currently, if she had claimed upon reaching SPA.  Improving her N.I. record would yield her an additional £75.84/week, so that she should receive £202.24/week once the payment to HMRC clears.  She will also receive ~35% extra for having deferred, depending upon when she starts claiming and how long she backdates. I understand that improving one's N.I. record after claiming is not applied retroactively to SP payments made before the funds cleared HMRC's account(s).  However, what I don't understand is: 1) If she is receiving a 35% uplift upon her SP from deferring and we update the N.I. record AFTER she starts claiming her pension, will the 35% uplift also affect the additional £75.84 she starts receiving once she improves her N.I. record? 2) What about if she waits until her N.I. record is improved before claiming her pension? 3) Finally, what if she waits until her N.I. record is improved before claiming but backdates the maximum 12 months to get a lump sum? This stuff is a minefield and we would appreciate any help anyone can give.  I've already asked the DWP, The Pension Service and MoneyHelper and received conflicting answers of limited degrees of certainty. Mostly, the typical places people are pointed to for pensions advice, say they don't know and refer me to another service that responds in kind and refers me back to them.

Hi,
Please contact The Pension Service as they are able to help people over UK State Pension age who have a query about their State Pension entitlement and about paying any voluntary National Insurance contributions to improve it.
Contact the Pension Service
If you prefer to write to The Pension Service then their address is:  
The Pension Service,
Post Handling Site A,
Wolverhampton,
WV98 1AF
Thank you.