Can a skin cancer lump leak small amounts of blood and pus?

Skin cancer : Rarely it can, mostly sign of an infection like a carbuncle, folliculitis, skin abscess or pemphigoid.

Unusual: This sounds more like a sebaceous cyst then it does skin cancer. Certain types of skin cancers when they get large can get infected and can have drainage of blood and pus. You should have a complete physical exam to figure out the reason for this drainage.

It is possible: It is possible. Best to bring this lesion to the attention of your friendly dermatologist.