If you're using Spectra, like on a Hennessy, then the sheath's purpose is to protect the Spectra strands on the inside. If those are exposed, I'd stop using the rings. You can put a new sheath on it to protect it, or if you don't stress the sheath anymore then you might be able to use duct tape for a temporary fix.
It's really not hard to replace the cords on a Hennessy, though...that's what I'd recommend.
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