Repair of peeling veneered board?

I've got some squares where the edge is peeling up.  Probably from little moisture now and then.  The veneer seems pretty thin, so I think sanding it would not be good.  It seems like it needs some glue under the lifted edges, but Don't know how to do that without lifting and probably cracking and messing it up more.  I'm not much of a wood worker type.  

Thanks.  You mean syringe with needle to get under it?  Will wood glue go through a hypo needle?

I was wondering if I could just smear some glue over the top, work it so that some of the glue gets in and under the veneer, and then wipe off the excess on the surface?  

much depends on the diameter of the needle, most woodworking glue is too thick for the medical kind.  Hobby shops sell syringes for glue.  Putting glue on top and hoping it will soak through in some fashion probably won't work, not enough will end up underneath, in any case you will need to clean the glue off the top, and how well that works depends on the glue.  Traditional hide glue used in veneering was put on both sides and wipes off (enough) with a damp cloth and takes refinishing.  PVA glue can be cleaned up but you can't really refinish over the top of it.  Things like cyanoacrylate (super glue) or epoxy do not clean up very well at all.  Better to get the glue underneath and minimise the squeeze out.  

Steve, how can a person worry over such a thing!  The Chinese call that stuff Wabi Sabi, evidence of usage and hey man that's a dope patina.