Click on the line between the blue arrows, using the Node tool. (This selects 2 nodes.) Click 'Delete segment between 2 non-endpoint nodes' button on the Node
tool control bar.
Select the 2 end nodes (at the red arrows) that you want to join, and click either 'Join selected nodes' button or 'Join selected end nodes with a new segment' button.
Note that only end nodes can be connected. It's not possible to connect nodes of a closed path, or nodes which are not at the ends of an open path.
I'm not clear exactly what you're asking. So if the above isn't it, you can explain a little more?