Refinishing depends on exactly what the finish actually is.
Lacquer can be re-disolved and smoothed out with lacquer thinner, and shellac with denatured alcohol.
Varnish hardens by polymerization and there is typically no solvent that can reverse this process.
Constantines sells small pieces of veneer and you may be able to find repair pieces there.
Depending on how well you want to hide the repair you can use anything from markers to paint to blend in newer wood to match the surrounding wood.

Using shellac to re-finish is often the best alternative, even though it is not water reisistant. It will apply over most old finishes and adhere.

What you are describing and brickeyee refers to is usually called "reamalgamation." It can be successful. In addition to the solvents mentioned, you can use an ATM stripper (Formby's comes to mind) See here But read the below article to see if you can get by without even doing that much.

Re-veneering is possible but likely you will find new veneers thinner than old veneers.

Here is a link that might be useful: Saving the finish

The "reamalgamation" method is often a failure on actual varnish.
Once polymerized these materials wil not dissolve and then harden in anything like the same manner they behaved originally.
If the finish is lacquer or shellac there is no need to buy expensive 'special' solvents to do the job.
Denatured alchohol or lacquer thinner work just fine (and are expensive enough).

I am aware that reamalgamators work only on lacquer or shellac. Most say so right on the can. Fortunately, unless the piece has been refinished by a non-professional, they are the most likely finishes on a 1930s piece. After thorough cleaning, I often pad or spray on shellac or lacquer.

Why pay extra $$ for a shotgun mix when plain old lacquer thinner or denatured alcohol is about $12 a gallon?
Most if the mixes are at least twice that, and some even four times since it comes in quart containers.