This is a simple question with a not-so-simple answer. All body cells, including cancer cells, do require glucose (a form of sugar) for growth - you can think of glucose as your body's "fuel" in the same way that gasoline fuels your car: without it, the body cannot function.
All forms of carbohydrates (whether simple carbohydrates such as sugar or complex carbohydrates such as breads and other starches) are broken down into their basic (sugar) molecules and travel through the bloodstream to body cells. If someone were to eliminate all carbohydrate from their diet (something that is discouraged and would be very difficult to do even if it were recommended), the body would convert other nutrients (primarily stored carbohydrates found in the muscle and liver or stored body fat) into glucose in order to produce the necessary "fuel" for the cells.
Thus, it is neither possible, nor recommended that in individual deplete glucose entirely from the bloodstream. That said, any registered dietitian would advise that sweets and other foods with added sugars be eaten on occasion and in moderation. These foods are usually lacking in important vitamins, minerals, fiber and phytochemicals beneficial for promoting health and fighting disease. M.D. Anderson supports the recommendations of the American Cancer Society (ACS) to consume a plant-based diet. This includes a variety of foods that contain some amount of natural sugar and starch (fruits, vegetables, beans, nuts, whole grains). The cancer preventative and cancer-fighting properties of these plant foods are too beneficial for them to be omitted from the diet. ACS also recommends that sweets and other processed foods be limited. For more information, visit the ACS web site (www.cancer.org).