I'm new to inkscape, drawing items on a tablet for papercutting.

I have a circle. I also have two lines leading to the circle. I want to erase the small area of circle between the two lines so it is welded together instead of cutting the circle off. How on earth do I do this?!

The erase tool either erases the whole object or the part of the circle i want but then puts in a new line.

The node edit doesn't work as the circle drawn in inkscape doesn't seem to have nodes in that area.

Welcome aboard!

You can: add two or more closed paths together that has some area enclosed, by pressing Ctrl++.
With that you could make lines welded to the circle.

Other way, to use the node editor tool:
make sure those paths are intersecting,
then duplicate the circle, zoom in and add nodes close to the intersection points, then snap them to those -with the snap nodes to intersection points option being enabled-,
un-connect the nodes in between, then
combine the two lines with the circle, connect unconnected nodes at the intersection points, and delete unnecessary nodes.
Can be done in seconds, with some experience.

Thank you for the reply - I will try these!