Although most children with leukaemia are cured using drugs (chemotherapy) alone, for some children additional treatments are needed. Stem cell transplant can cure children where chemotherapy and other drugs have failed. In this case, the immune cells of the donor attack the leukaemia cells of the patient. Cord blood collected from the placenta of unrelated babies is often used as a donor cell source and appears to work well at controlling leukaemia and less likely to cause complications such as when the immune cells also mistakenly attack healthy tissues (called graft versus host disease, GVHD). The investigators have noticed that during cord blood transplant, the donor immune system appears to recover more quickly and not be associated with GVHD, when a type of blood transfusion containing white cells are also given to the patient. The infused white cells appear to stimulate the donor immune cells to expand much more than usually seen. During this research, the investigators will study this immune cell expansion during cord blood transplant in children with difficult-to-cure leukaemia who also receive a transfusion of white cells, termed granulocytes. The investigators will assess the safety of the effects of the white cell transfusions and the immune cell expansion on the child, and look at the outcomes on the patient's leukaemia, and whether there is GVHD or not.