Yeah, in that case you can just replace that wire. If you don't have any dead flybacks, maybe try to find some soft HV wire of a similar gauge. It will have to be the same thickness, if you want to keep the old cup, as it matters in that case. But finding a dead flyback and salvaging the cup and wire would be easier.
The white ferrule on the wall of the cage unscrews, and you can take that whole assembly apart, and repair it. You'd just have to remove the metal cup from the diode end of the wire, and solder it on to a new wire (with the boot and spring, etc).
If you want me to do it for you, I can. I've done them before, and have the parts.