Yes, phagocytes are part of your immune system and immunologic response. They help to protect us from disease. Phagocytes are a type of leukocyte as is a neutrophil. Phagocytes search out and engulf and destroy foreign invaders in the blood like bacteria, viruses, and fungi.
Yes, there are many microorganisms that the phagocytes can't engulf and destroy. Cancer cells are a good example. Most types of cancer are transmitted through the lymphatic system not the bloodstream. Certain leukemia's are susceptible to phagocytosis but the number of immature leukocytes seen in leukemia is so great that the phagocytes can not destroy enough to make any appreciable change in the patients condition.