I glue alot of "broken" pieces with "Titebond" and clean it up without a problem. If it has a gap, I mix sawdust with shellac or pollyurethane or water based wood filler and fill the hole.
Trust me, YOU are the only person to notice it!! Especially if it is a fret type item, no one will see it. And if they do, SO WHAT!?
The true test of a good woodworker is the ability of corecting OOPPSSEEESS which happen to all of us at one time or another.
Rog