I would suggest that you take some other photographs printed on similar paper, and experiment. Sweat onto them, let it dry, then try different techniques.
A: Put a puddle of distilled water on a stain, swirl with a Qtip for N minutes, then pull the water off initially with a paper towel, and finish with a piece of lens tissue. (The water wicking into the paper towel won't shed onto the print.
B: Same idea, use distilled water mixed 200:1 with photoflo.
C: Start with 1 liter of warm water and 2 drops of uncoloured dish detergent. Swirl, blot, and immediately rinse with B, above. The idea is that detergent will remove residual skin oils.
D: If any of the above worked, try repeating it. If it worked slowly,wet it repeatedly, then dry it by putting down a piece of lens tissue backed up with a paper towel. (A towel by itself will leave towel lint embedded in the gelatine.)
E: is the stain lighter than the print? If so you may be able to use retouching colours to fix it. Get lots of practice first. It's an acquired art.
Examine with a 10 power jewelers loup. You are looking for the following:
is there an actual change of colour between inside the ring and outside the ring?
is all the discolouration just in the ring itself? (Surface tension favours junk collecting at the point where the liquid meets the solid surface. Coffee stains are rings, not spots.
is there a height difference between inside and outside? (The gelatine has swollen and not shrunk back. You may want to use a very low angle illumination to check this. An LED flashlight laying on the table works well.
is there stuff stuck to the print?
After working on it, do you still have a stain? Have you moved the stain? If it's a ring stain, not a spot stain, can you move the edge of the stain to an inconspicuous location?