I think that cutting off and and retying is a good thing to do. You don't need to wind it back and rethread, just tie on again and be consistent when tensioning the knots. Also see if you can ease up on the warp tension a bit overall while you are weaving.
I want to check on one thing - do you have packing on the warp beam that separates the layers of wound warp from each other? And does that packing extend at least a few inches beyond the warp threads at both ends? Packing is important for maintaining even tension across the width of the warp. If you have it, great, that's one less thing to worry about.
If it is always the same threads in the same place, it may indicate a sharp bit or burr in either the heddles (if metal) or the dent of the reed they go through.
If it is only at the very edge, you might have better luck switching to a stronger thread in a similar colour, like a silk or a cotton in a similar weight. Worst case scenario, sley a strong fishing line one dent out from the edge, weighted off the back, and use that as a protective measure for the selvedge threads. But hopefully cutting off and retying on plus easing up on the warp tension a bit will handle the issue. This paragraph is kind of last ditch effort stuff.
Let me know if any of this helps. These are all things that have helped me at one point or another, in my weaving explorations. I hope your scarf two goes well!
