This phase I trial studies the side effects and best dose of BSB-2002 when given after cyclophosphamide and fludarabine and tests how well it works in treating NPM1-mutated acute myeloid leukemia (AML) that has come back after a period of improvement (relapsed) or that does not respond to treatment (refractory). BSB-2002 is a type of personalized autologous T cell receptor-modified T cell therapy. T cells are infection fighting blood cells that can kill cancer cells. The T cells given in this study come from the patient and have a new gene put in them that makes them able to recognize mutated NPM1, a protein on the surface of cancer cells. These NPM1 mutated-specific T cells may help the body's immune system identify and kill NPM1-mutated AML cells. Giving chemotherapy, such as cyclophosphamide and fludarabine, before BSB-2002 helps kill cancer cells in the body and helps make room in the patient's bone marrow for new blood-forming cells (stem cells) to grow. Giving BSB-2002 after cyclophosphamide and fludarabine may be safe, tolerable, and/or effective in treating relapsed or refractory AML in patients with NPM1 mutation.