There is a shape kinda hidden in the design. If you press Ctrl+6 it will open this view where you can pull the border around to see everything reduced to black strokes. If you dont want that hidden object select it and delete it.
The way i would do what you want to do is to cut the one clue circle form the other then add in the red circle to the ring you make.
select both blue lines and select difference
It will look the same but you have actually cut the center out of the shape. if you add a fill you can see it.
Now click the blue ring and the red circle and select union
Then it should be what you are looking for.
If you want to cut the inner part before the outside select the part and select break apart
select the inner line and assign it a different color.