The purpose of this research study is to examine the survival of patients undergoing partially matched hematopoietic stem cell transplant (HSCT) on a new type of treatment approach, which has been developed specifically for patients who have evidence of their disease at the time of transplant. In this research study, a way of strengthening the response of the donor cells against the disease has been developed. Patients will undergo one additional day between the two steps of the transplant which may allow their donor's cells to fight the disease more effectively.