Is there a way of fixing or touching up damage to varnished wooden surfaces?

Depends on the mark.
While I am not well versed on different finishes my Father restores antiques and passed the following on to me:

If it is white and cloudy then you can take regular fireplace ash and mix a paste with a tiny bit of water and rub it till it clears up.

To refloat the finish I would advise some restoring finish like Howards which is the easiest way to go.

It was years ago, but my husband had good results using Formby furniture refinisher to redo a walnut framed mirror that belonged to my grandparents.
http://www.formbys.com/products/
It liquified the opaque old varnish and left it looking good without stripping or sanding the wood. For your purposes, it sounds like the facelift kit might work.

I would second @MagsRags suggestion of Formby’s. Also, if you’re in a US city, you may want to check and see if there’s a Furniture Medic franchise in your community. This is a service that comes to your home and fixes minor damage to furniture on the spot. I’ve used them for spilled perfume before.

If you have a water ring on wood furniture, rubbing mayonnaise into the ring will usually remove it.

The Formby’s Refinisher looks like it’s designed to actually remove varnish, though. They make it sound like a prep for applying new varnish instead of restoring the old varnish.

The best repair procedure will depend on the nature of the original finish. If it is a modern piece of commercially produced furniture (as opposed to an antique), then there’s a good chance that the finish is a nitro-cellulose lacquer. That would also fit in with the fact that the alcohol in the perfume dissolved it (shellac will also dissolve in alcohol, but it’s rarely used in commercial furniture).

If that’s the case, lacquer is relatively easy to spot repair (which is to say still hard, but easier than other finishes) because new lacquer blends itself in well with old lacquer. You’d need to rub out the damaged spot with a fine abrasive (I’d never heard of @Dog ‘s ash trick before, but it sounds worth a try). Normally a pumice powder and a bit of oil would be used. Then a few coats of a spray nitro lacquer, followed by sanding with progressively finer abrasives to level and polish the repair. It’s a lot of fussy work to make a perfect spot repair.

@harp, some of the varnish was removed, which in our case was what we wanted. Because it all liquified, he was able to spread it all around with his rubbing cloth and end up with a thinner but uniform coat that looked much better than he started with. He didn’t have to apply any new varnish.

But you’re right, that was an antique, and it might not work the same on a newer piece. If you were going to try it, you’d probably want to do a test patch on the underside.

Here’s what I did in the end:

2x coats of linseed oil with tint
Once dried, painted small damaged areas with clear nail polish to replicate the glossy finish.

Certainly isn’t perfect but it looks much better than it did when it was first damaged.