Like @spikeyxxx , the skin modifier is rarely used by me. When I have used it there is always some issues to deal with.
From what I understand is the root needs to be at the start of the chain of points. It is quite often used when modelling trees, and the root would be , well at the root of the tree and grow from there and branch out. The modifier can get confused if the root is not in the right place.
Another common issue is if you have a branch at a tight angle you may find a similar behaviour.
There are lots of issues with this modifier, it takes a lot of trial and error to get the hang of it.