I think copper-toned prints are quite liable to marks, although I've never had such a problem - I always use tongs and handle only the edges of the print. Touching the image area itself isn't a good idea, and can make the toning patchy.
If the 'marks' you mention are like a blooming on the surface, this is a normal part of the copper-toning process and can be removed by carefully wiping with (wet) cotton wool whilst rinsing.
If the print isn't fixed properly, or washed properly, you'll also get marks, I think you do have to be a bit more careful even than with sepia & other toners, but it sounds to me like it's quite likely to be contamination from the gloves, during toning .....
Hope this helps in some way!
Cate