Can I top up Class 2 NI contributions to qualify for ESA?

I applied for ESA last week. They called me today to tell me my NI contributions were insufficient for the years needed, but they can go ahead with contributing NI credits instead. I agreed to do this. I have been registered as Self-Employed for the past few years although I'm not earning much at all (less than £2000). I thought I was going to qualify for NI contributions via some standard employment I did, so never even took my Self Employment into consideration until after the phonecall. I think I was looking at the wrong years. I have never paid Class 2 NI contributions on my self-assessments so far - but, am I correct in thinking I am able to 'top-up' previous years now? Two questions if yes: 1. How much would this cost? 2. If I did this, would ESA be able to re-process my claim ASAP (so that I can receive the benefits instead of the credits)? Or will I have to wait a certain amount of time? Thank you.

Hi,
The option to pay Class 2 NICs voluntarily is outlined on your tax return, the relevant option should be selected and the payment made through SA before 31 January following the end of the tax year.
It is now too late to pay Class 2 NICs voluntarily through SA, you will need to call the NI helpline on 0300 200 3500. Lines are open 8am – 6pm Monday to Friday.  
They will confirm the relevant rate and  how much is needed for each year you wish to pay and how to make payment.  
Anything paid after  31 January following the end of the tax year will be marked as ‘late paid’ and this will potentially affect how the NICs are used when assessing/re-assessing any claim to benefit.
Please note that the NI helpline cannot give any advice on benefit entitlement so you will need to know which years you wish to pay before calling. Any enquiry regarding benefit entitlement and how the late payment of NICs will affect a claim to ESA will need to be directed to the DWP
Thank you.