As long as the phone has developed a fault and has not been damaged, I would have thought that it's Tesco's responsibility - as long as you're still in your contract.
I'm only going off my experience with Three. My phone was 12 months old ish and I was on an 18 month contract. My phone developed a fault (can't remember what it was) - I sent it back to Three and they repaired it - all F.O.C. including the postage, 'cos if I remember correctly they sent me a pre-paid envelope to post the phone in.
Most insurances are mainly for loss of phone and accidental damage. I'm hopefully not tempting fate here, but I've never taken insurance out, even though I've had some decent phones - I really can't see the point, but that's just my opinion.