How can I fix the veneer on this cabinet?

It looks like you may be able to disguise it. I would go into your local hardware store and get a look at the stain options. You may be able to find a stain (or mix a couple of them if need be) that will match the finish so you can stain the places where the veneer is missing. That would help make it a lot less noticeable.

I would use something like Howards refinishing stain. That way you don't have to sand anything.

I'm not sure exactly what you mean. Taking off the veneer or removing the finish. I refinished a veneer corner desk recently and fixed the veneer with wood Bondo. I left the nice parts of the veneer wood natural and painted the repaired areas. You can see the tutorial on my blog. Hope this helps!

What I would do. Sand the edges of the veneer with 220 grit sandpaper feathering into the damage area to smooth it out. Then I would get some wood stain or a few to closely match. Might need to mix need to mix a few. First try on a piece of sample wood to get a close color tone.

Go to Lowe's or Home Depot and buy Rejuvenate which is sold in the stain section. They have wood repair markers about 6-7 to a package with color indicators on top of the cap. I would think perhaps blending 1-2 of the colors onto a small test area might blend in with the existing cabinet and will be much easier than trying to restain the entire door.

I would try a wood filler product (as described above) to level out the missing pieces of veneer. Remember this is a band-aid, not a total re-furb. Patch the "holes" and sand them level with the original surface. You're going to need fine grit sandpaper - at least 220 as mentioned by William and you might even move to a higher number grit depending on what the original surface is like. Most of these fillers take the stain a few shades darker that the wood - so be aware of that. I'm an artist and personally, I wouldn't use stain but do a wet blend with acrylic paint and then cover that with a gloss finish - or whatever the cabinet's finish is. No one will ever see the patch!

Hi! I agree with the above that you should fill in the area with some sort of wood putty. If you just try to stain it, you will notice the gaps where the veneer is missing. But, before you do that, make sure you glue down the edges of the existing veneer so that they are firmly attached. Otherwise, they will continue to peel off and you won't get a smooth transition where you add the putty.

I would just send all off and re- coat with new stain or paint