Try to find out if it’s a lacquer finish, or whether it’s something like a Danish Oil. If you Google those terms, you’ll see some images that should help you decide what it is.
If you’ve damaged lacquer, that’s a professional job to put right. If it’s more of an oil/varnish finish, then you might be able to do a patch repair using one of those wax sticks in a matching tone.
Otherwise, I think you’re looking at a strip, sand and re-oil.