Can I buy apple care for a second hade macbook air?

I have recently bought a Macbook air from someone on Gum Tree, we met close to a coffee shop and I gave him £500 cash which I thought was a fantastic deal. When I start using it properly, intermitantly the keyboad and trackpad become frozen. Sometimes it stays frozen.

Today i took it to the Apple centre on regent street, we started it up fine but froze so the clever mac guy tried a usb keyboard and mouse an it worked without problem, So we identified the problem is with the upper casing,(trackpad and keyboard) this would cost about £120 to replace.

I have cocerns that if i fix this then its another issue, I am left with endless bills.

Can I purchase apple care on my new secondhand laptop and get it repaiered under the new cover?

Other info : I have spoke to the guy who sold it to me 11 days ago and have been trying to get my moneyback, he says it was fine when he sold it to me. I know it wasnt.

Hope someone can help.

Thanks, Jon