When I received the painting, I was happy to see that the areas to be repaired were small. Also, I'd noted on the back of the painting the type of varnish I'd used, which allowed me to pick out the right solvent for removing it. If you use damar varnish, the varnish is soluble in turpentine; it does not dissolve in odorless mineral spirits. Fortunately, I'd used Gamvar, Gamblin's arylic resin varnish, which does dissolve in OMS, and that made the job easier.
Using cotton balls, I used a little Gamsol (OMS) and a delicate, circular motion to remove the varnish from the entire painting. You can't just remove one area, as it is nearly impossible to get the varnish in that area to match the rest of the painting when you re-varnish. (I'm glad this wasn't a big painting!) What's interesting is that, even with a delicate touch, I picked up a little color on the cotton.
Once the OMS had dried off, I applied retouch varnish (Gamvar diluted 50% with Gamsol), which made the painting look "wet" again so I could match color better. I was able to match the paint exactly. I applied it with a small brush, taking care to "feather" the color around the spot and over the undamaged area so the spot would blend and disappear.
The only problem I ran into is that in one location the underlying gesso had been damaged, creating a groove. I was able to fill this in somewhat with thin paint, but if the light is right, one can still see a very slight depression. One might have tried filling in the groove with gesso first, but since this was a traditional gesso (made with rabbit skin glue and chalk) and not acrylic, I didn't want to step into the dangerous territory of trying to get a patch to hold. The groove is hardly noticeable now.
Although I am very happy with this repair, it's not something I want to make a career of. Please be careful with those paintings!