Is it safe to take hydrocodone to reduce the pain and tenderness of mono?

Yes. Yes it will. However, take sparingly as most hydrocodone preparations are combined with Acetaminophen which can "stress" the liver. Liver inflammation is present in mononucleosis.
The Moans of Mono. Most viral syndromes, such as infectious mononucleosis, have associated myalgias (muscle aches) and arthralgias (joint pain). It is not well understood why this occurs but is believed to be due to various immunologic responses in the body. A opioid such as hydrocodone is overkill for such discomfort. Most often a warm bath, cool shower, or massage helps. Tylenol (acetaminophen) is safe. Sleep is a must.