Does pus normally come out of cancer lumps?

No: Most cancers grow as solid tumors. On occasion, a tumor can grow very rapidly and outgrow it's blood supply. The center of the tumor can then liquify; this typically will contain bloody fluid rather than pus.

It can: Cancerous growths that are growing rapidly or have out grown their blood supply can become necrotic. Necrotic tissue can have a purulent discharge or pus. This is because it is dead or dying tissue that can become infected. As an example, a breast abscess that is being drained, needs to be biopsied to make sure there is not an underlying breast cancer.