This phase I trial studies the side effects and the best dose of genetically modified T-cells after cyclophosphamide in treating patients with relapsed or refractory acute myeloid leukemia. An immune cell is a type of blood cell that can recognize and kill abnormal cells in the body. The acute myeloid leukemia-specific immune cell product will be made from patients' blood cells. The immune cells are changed by inserting additional pieces of deoxyribonucleic acid (DNA) (genetic material) into the cell to make it recognize and kill acute myeloid leukemia cells. Placing a modified gene into white blood cells may help the body build an immune response to kill cancer cells.