You'd be best to buy a new adapter in the meantime. Once it starts to fray, it only gets worse very fast. The woven shield wire that you see is the ground path for the power (bad design IMO). It will get hot, causing even more wear.
In the meantime, you could do this (at your own risk) ...
* carefully and cleanly strip back a bit of the insulation on the thinner wire leading from the fray to expose a bit more of the woven shield wire - goal: open a good path to reconnect across the frayed shield wire
* tightly wrap just the woven fray with aluminum foil to overlap completely through the fray - goal: reconnect the two frayed ends of the woven shield
* tightly wrap that part with something insulating (electrical tape, electrical putty, epoxy ...) - goal: re-insulate the frayed part
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JJW