It would depend what the print was created with. If the print was created with some kind of ink jet system, there is no way to clean it. The inks are soluble, and pretty much any cleaning agent will ruin the ink. The adhesive itself has also likely bonded to the ink particles, so trying to remove the adhesive will damage the print.
Some forms of print are more resilient, such as laser or possibly dye sublimation. Generally speaking, I would assume that there are few cleaning agents that would allow you to remove the adhesive without also removing the ink or ruining the paper substrate.
When it comes to board-mounted prints, usually after bonding the print to the base, you seal the print with some kind of spray-on sealant or laminate. If you use a print sealer, it might help mitigate the effects of the adhesive, as well as make your print resilient to any future damage.