What could be the underlying illness of cuts on my foreskin?

Balanitis? It may be that your foreskin is just too tight; but it could also be balanitis, a skin condition affecting the head of the penis. This is not necessarily a sexually transmitted disease. It may be due to a fungal, bacterial, or yeast infection.Wash the area right after sexual activity with a mild soap and apply a moisturizing cream. If the problem persists, see your doctor.
Probably phimosis. Phimosis or tight foreskin can result in minor splits in the foreskin whin it is retracted, these may mimic "cuts". You may require circumcision, or steroid cream. Consult with a urologist.