I meant to mention this in my post this morning, but I forgot. I wish I had before and after photos to illustrate this, but I didn't take any before we started as I wasn't at all sure it would work.
While the builders were here they managed to tear the vinyl flooring in the kitchen, which we weren't keen to replace as it is only about eighteen months old. I bought a kit online, of which I was highly suspicious. But it worked. We painted on the magic gunk (went with clear as we thought a rubbish colour match would be more noticeable than a tear) and then heated it. We abandoned the silly little 'heating tool' they provided, which was a little lump of metal on a stick that you were supposed to heat with an iron. We just used the iron itself, with a cloth underneath.
It worked very well indeed and I'm happy to live with the floor as it is now. It isn't exactly pristine anyway, owing to an unfortunate incident with a permanent pen and a child.
Here's the mended bits. There were two bigish tears - about five cm each.