It's called rim tape and, yes, you should certainly move it back over the spoke hole inside the rim as it's there to prevent just what happened to your tube. Better yet, but a roll (or two) of Velox rim tape, which is adhesive backed cloth and the world's standard for rim tape. Remove your unreliable thin plastic one and replace it with Velox and, applied properly, it will last as long as the rim. Velox comes in three widths so ask your LBS which one is proper for your rim.