dealing with reflections on the image of an oil painting

[Newmarket2]

When allowed (and quite a few museums do allow non-flash photography), I've been taking pictures of my favorites works.
To varying degrees, almost every shot of an oil painting has some noticeable reflection off the shiny surface. In some cases, where there is a spotlight, the glare is severe, but in some cases, it's modest.
I've been fooling around with trying to deal with the problem in LR (I'm sure there are more tools in PS)...largely adjusting down the highlights sliders.
I was wondering if anyone had worked their way through this already and had some advice?

 

[Nogo]

Either add contrast or clarity. Either one will help with the glare. Just have to figure out the best balance between less glare and too much clarity or contrast. There is no way to correct it perfectly, you just have to balance out the results the best you can.

The other thing you can do is use a polarized filter on the camera and stop the glare in the first place. If you do that, you will have to correct the white balance more than you would without the filter, but you should be able to recreate the work of art much closer to reality than anything you can accomplish in Lightroom.