Life is too short to fool with something like that unless you see the
crack as a "feature". I used to screw with things like that a lot (and
still do if I want) until as Darrell pointed out, the glue will
contaminate and discolor your finishing in many cases.

CA glue doesn't hold up well in heat. So if you are using a friction
finish and really "burn it on" you will be guaranteed problems.

I have also had CA pick up (especially the water thin used repeatedly
in a crack) dark stuff that lives in the crack and make itself apparent
with other finishes.

And no one needs a piece of wood flying off the lathe this close to
Christmas; if it is a broken up as you say it is, why take a chance?

Robert