How do I repair a table shelf that has layered veneer?

I wish otherwise - but this is usually close to impossible to fix well. If you plan to paint it it’s easier - you can replace the whole section with wood and cover with paint.

I agree with the top answer, once veneer splits like that there is very little you can do but replace it. If you're painting you can peal the majority of it off, fix with wood putty, sand down and then paint for a smooth finish that will hide the water marks.

Not much you can do other than replacing

Replace it. No way to repair.

Hello you can remove the damaged veneer or cover it up.

Hi! I have had little success with trying to glue veneer back down. The good thing is that there is often decent wood under veneer on older pieces of furniture. If the water damage hasn't ruined it for stain, you may able to refinish the underlying wood. I have found some pieces can look really good with stain. Others, not so much. Sometimes all the sanding and stripping in the world won't fix it. Then, if a gel stain won't fix it, I look at painting it. You can also buy veneer to apply, but, it isn't cheap. What are you going to use that shelf for? If it's going to be covered with books or plants? Removing the veneer and cleaning it up may be the easiest answer. Good luck!

found a couple ideas for you

https://salvagedinspirations.com/how-to-remove-veneer/

https://home.howstuffworks.com/home-improvement/home-diy/projects/how-to-repair-wooden-furniture-veneer.htm

There's a discussion about using a heat gun in this post. I don't know if it's applicable to your situation, but it's worth a read. https://www.hometalk.com/diy/repair/furniture/q-can-bubbled-veneer-be-fixed-10557904

Your best bet is to remove the veneer and replace it or just finish the wood beneath.

Hello, I would just cut new piece of wood and paint it all one color, less work than trying to sand and fill the veneer damage, chalk paint is very forgiving and would look awesome on it.

https://www.youtube.com/watch?v=d60p3n79uEo

Remove the veneer, and replace the wood, unless you are painting the piece. You do not need to replace the wood after sanding.

You can remove the damaged veneer and replace it with a new piece. Cobbling together repairs is a difficult task, it’s better to remove and replace.

You should be able to pull off the buckled veneer and there will be nice wood under it

Thank you everyone for your help. I believe I have enough suggestions so I can give some new life to this little table.

How to Repair Wooden Furniture Veneer

https://home.howstuffworks.com/home-improvement/home-diy/projects/how-to-repair-wooden-furniture-veneer.htm

There are lots of projects here that can help you: Home and Garden DIY Ideas | Hometalk