Actually I thought of a better way!
As far as I can tell, without the SVG file, those 8 short paths are subpaths of a compound path. Do Path menu > Break Apart, to make them all separate paths.
Select each of the 2 short paths which you need to sort of cut the circle with, one at a time, with Selection tool. While you're holding Ctrl and Shift, grab a corner arrow, and scale them just until they completely cross the circle. (You can't scale them together, because they might be slightly displaced.) After they're both crossing the circle, select them both and do Object menu > Raise to Top (or this button on Selection tool control bar
)
While they're still selected, do Path menu > Combine (makes them 2 subpaths now).
Select the circle and the 2 short paths, and do Path menu > Cut Path. It might look like nothing happened, but in fact, the circle has been cut.
Now you should be able to select the arc and delete it. (Note that the 2 short paths are sort of consumed in this operation, so if you need them later, be sure to Duplicate them before you start.)
As I said, this way makes it very precise. If you don't need precision, you could just add the nodes to the circle path manually, and use Delete Segment Between 2 Non-End Point Nodes. Note that you do not have to do Break Path at Selected Nodes first.
