I have a chest that has a few slight scratches and also has two deep scratches that went through the finish. Can I remove these without completely removing the finish and starting over?
No. Unless the finish is lacquer---which can be repaired by adding more lacquer---because lacquer tends to disolve the previous coat of lacquer and flow out smooth.
The way to find out if a finish is lacquer is to wipe a small area with lacquer thinner----lacquer is a finish that can be disolved by its thinner.
Most other finishes have to be sanded or removed to repair.
Should I spray or brush it on?
Just wet a cloth with the lacquer thinner and blot it on an inconspicuous area. You will be able to tell if it is softening the finish.