Unfortunately if you have simply failed to claim anything over the past 10 years, even with good cause, you cannot simply go back and claim it now. So I don't think it would be possible to make a credits claim for that period.
If you're 60 though, it's worth checking your entire working life to make sure all the years of work are accounted for. If you're willing to set up an online account with hmrc, you can check this online https://www.gov.uk/check-national-insurance-record
Aside from this - there are three points I would like to make to you. Firstly, that even if your contribution record is not complete, you will still receive State Pension, it will just be at a reduced rate - again you can follow the link above to see how much you will receive; Secondly, if your income on State Pension is below the amount the government thinks you need to survive on you can claim Pension Credit as a top-up (which is like Income Support for pensioners); and thirdly, at the present time, you could still make a claim for ESA (credits only if household income/capital is too high) and possibly receive 5 years credits before State Retirement age depending on your date of birth.
Hope this helps. Let us know if you want to know anything more on this.