I would use superglue, it is thin, it is not for filling gaps but it sound as you do not have a gap. The wood need to be dry to give the glue a chance to flow in the gap. If you go to a model / hobby store they have gap filling superglue but I think you would have trouble getting it in the gap.
The other thing I would think about is to thin varnish and paint the gap until it is filled.