BSB-2002 After Cyclophosphamide and Fludarabine for the Treatment of Relapsed or Refractory Acute Myeloid Leukemia Patients With NPM1 Mutation

Inclusion Criteria: * Documented informed consent of the participant * Note: For research participants who do not speak English, a short form consent may be used with a City of Hope (COH) certified interpreter/translator to proceed with screening, while the request for a translated full consent is processed * Age: ≥ 18 years * HLA-A\*02:01 * Eastern Cooperative Oncology Group (ECOG) ≤ 2 * Adequate venous access for apheresis or agree to use of a central line for apheresis collection * AML diagnosed per ELN criteria which has been treated with at least two lines of therapy, and meet one of these 3 criteria: * Which is relapsed (after previously complete remission, CR, CRh or CRi), OR * Are refractory (failed to achieve complete remission) to the last treatment * Primary refractory patients should have received at least two cycles of induction treatment, OR * MRD positive (at least 1% leukemic blasts in blood or bone marrow) after being MRD negative following the last treatment * Positive for NPM1 mutation type A, D, G or H. The confirmation for NPM1 mutation must be performed by NGS within 3 months from enrollment * If participant has had prior hematopoietic cell transplant (HCT), all 3 of the following must be met: * More than 3 months from transplant at the time of enrollment * No clinically significant graft-versus (vs)-host disease requiring systemic treatment * Any non-hematological toxicity related to transplant has resolved to \< grade 2 per Common Terminology Criteria for Adverse Events (CTCAE) * Total bilirubin ≤ 2 X upper limit of normal (ULN) (unless has Gilbert's disease or related to leukemia involving the liver) (performed within 6 weeks prior to leukapheresis unless otherwise stated) * Aspartate aminotransferase (AST) ≤ 3.0 x ULN (unless related to leukemia involving the liver) (performed within 6 weeks prior to leukapheresis unless otherwise stated) * Alanine aminotransferase (ALT) ≤ 3.0 x ULN (unless related to leukemia involving the liver) (performed within 6 weeks prior to leukapheresis unless otherwise stated) * Creatinine clearance of ≥ 40 mL/min per 24 hour urine test or the Cockcroft-Gault formula, or serum creatinine ≤ 1.6mg/dL and the participant is not on hemodialysis (performed within 6 weeks prior to leukapheresis unless otherwise stated) * No history of congestive heart failure class III or IV New York Heart Association (NYHA) OR left ventricular ejection fraction (LVEF) ≥ 45% up to 90 days before enrollment * If able to perform pulmonary function tests: Forced expiratory volume in 1 second (FEV1), forced vital capacity (FVC) and diffusion capacity of the lung for carbon monoxide (DLCO) (diffusion capacity) ≥ 50% of predicted (corrected for hemoglobin) * Note To be performed up to 90 days before enrollment * If unable to perform pulmonary function tests: Oxygen (O2) saturation \> 92% on room air * Note To be performed up to 90 days before enrollment * Seronegative for HIV antigen (Ag)/antibody (Ab) combo and no active hepatitis C virus (HCV) and hepatitis B virus (HBV) (surface antigen negative) (performed within 6 weeks prior to leukapheresis unless otherwise stated) * If seropositive for HIV, HCV or HBV, nucleic acid quantitation must be performed. Viral load must be undetectable * Meets other institutional and federal requirements for infectious disease titer requirements * Note Infectious disease testing to be performed within 28 days prior to leukapheresis * Women of childbearing potential (WOCBP): Negative urine or serum pregnancy test (performed within 6 weeks prior to leukapheresis unless otherwise stated) * If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Agreement by females and males of childbearing potential to use an effective method of birth control or abstain from heterosexual activity for the course of the study through at least 12 months from the date of BSB-2002 infusion * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) * CRITERIA TO PROCEED WITH LEUKAPHERESIS: Research participant has signed the informed consent * CRITERIA TO PROCEED WITH LEUKAPHERESIS: Research participant must have appropriate venous access, have a central line or be willing to undergo central or temporary line placement * CRITERIA TO PROCEED WITH LEUKAPHERESIS: Biochemistry laboratory results have been reviewed for clinical significance and appropriate measures taken. Hematology assessments are expected to be out of the normal range and do not need to be assessed for clinical significance * CRITERIA TO PROCEED WITH LEUKAPHERESIS: The last dose of systemic chemotherapy must be at least 2 weeks or 5 half-lives, whichever is shorter, before the leukapheresis procedure with the following exceptions: * Steroids and vincristine are allowed up to 7 days prior to leukapheresis * Intrathecal chemotherapy is allowed up to 3 days prior to leukapheresis * Hydroxyurea is allowed up to 48 hours prior to leukapheresis * The research participant cannot be on prednisone or equivalent doses of other corticosteroids at the time of leukapheresis. Note: Topical and inhaled corticosteroids in standard doses and physiologic replacement for subjects with adrenal insufficiency are allowed * CRITERIA TO PROCEED WITH LEUKAPHERESIS: The last dose of prior targeted agents, immunotherapy or radiation must be at least 2 weeks or 5 half-lives, whichever is shorter, before the leukapheresis procedure * CRITERIA TO PROCEED WITH LEUKAPHERESIS: If the research participant has undergone prior HCT, at least 3 months must have elapsed since receiving transplant to undergo peripheral blood mononuclear cell (PBMC) collection for BSB-2002 manufacturing * CRITERIA TO PROCEED WITH LYMPHODEPLETION: Research participant's BSB-2002 product is received by COH * CRITERIA TO PROCEED WITH LYMPHODEPLETION: Biochemistry laboratory results have been reviewed for clinical significance and appropriate measures taken. Hematology assessments are expected to be out of the normal range and do not need to be assessed for clinical significance * CRITERIA TO PROCEED WITH LYMPHODEPLETION: Research participant with no active CNS leukemia * CRITERIA TO PROCEED WITH LYMPHODEPLETION: Research participant must meet the following washout criteria: * Regimen: Hydroxyurea; Required washout period: Stopped prior to start of lymphodepletion * CRITERIA TO PROCEED WITH LYMPHODEPLETION: ECOG ≤ 2 * CRITERIA TO PROCEED WITH LYMPHODEPLETION: Women of childbearing potential (WOCBP): negative urine or serum pregnancy test within 30 days prior to the start of lymphodepletion. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * CRITERIA TO PROCEED WITH LYMPHODEPLETION: Research participants of reproductive potential must agree to use and utilize an adequate method of contraception throughout treatment and for at least 12 months after BSB-2002 T cell infusion * CRITERIA TO PROCEED WITH LYMPHODEPLETION: Not requiring supplemental oxygen or mechanical ventilation, oxygen saturation 92% or higher on room air * CRITERIA TO PROCEED WITH LYMPHODEPLETION: Not requiring pressor support, no symptomatic cardiac arrhythmias, no acute coronary syndrome, or uncontrolled hypertension * CRITERIA TO PROCEED WITH LYMPHODEPLETION: Calculated creatinine clearance (absolute value) of ≥ 40 mL/minute * CRITERIA TO PROCEED WITH LYMPHODEPLETION: Total bilirubin ≤ 2 times the institutional upper limit of normal (ULN) * Note: In the event a participant has elevated levels of liver enzymes possibly related to underlying disease/disease progression, the participant will still be considered eligible * CRITERIA TO PROCEED WITH LYMPHODEPLETION: ALT ≤ 3 times the institutional ULN * Note: In the event a participant has elevated levels of liver enzymes possibly related to underlying disease/disease progression, the participant will still be considered eligible * CRITERIA TO PROCEED WITH LYMPHODEPLETION: AST ≤ 3 times the institutional ULN * Note: In the event a participant has elevated levels of liver enzymes possibly related to underlying disease/disease progression, the participant will still be considered eligible * CRITERIA TO PROCEED WITH LYMPHODEPLETION: No new neurological deficits * CRITERIA TO PROCEED WITH LYMPHODEPLETION: No clinical evidence of uncontrolled active infectious process * CRITERIA TO PROCEED WITH LYMPHODEPLETION: Participants must be cytomegalovirus (CMV) negative (by polymerase chain reaction \[PCR\]) * CRITERIA TO PROCEED WITH BSB-2002 INFUSION: Biochemistry laboratory results have been reviewed for clinical significance and appropriate measures taken. Hematology assessments are expected to be out of the normal range and do not need to be assessed for clinical significance * CRITERIA TO PROCEED WITH BSB-2002 INFUSION: Prohibited medications have not been administered * CRITERIA TO PROCEED WITH BSB-2002 INFUSION: ECOG ≤ 2 * CRITERIA TO PROCEED WITH BSB-2002 INFUSION: Absolute lymphocyte count (ALC) \< 500/µL (0.5 × 10\^⁹/L) * Discuss with sponsor if ALC target is not achieved by day -1 * CRITERIA TO PROCEED WITH BSB-2002 INFUSION: Patient must not have any medical condition that render the patient unstable, including but not limited to untreated infection, altered mental status, unstable vital signs, worsening cardiovascular status, requiring discussion with the sponsor Exclusion Criteria: * Leukemic blast count of \> 20,000/µl. If the blast count can be maintained below the threshold with hydroxyurea, the patient would be eligible * Extramedullary only AML * Central nervous system (CNS) involvement refractory to intrathecal chemotherapy and/or standard cranial- spinal radiation * Candidates for hematopoietic cell transplant * Eligible to receive an approved targeted therapy * Treatment with other investigational agents within 5 half-lives of the planned dosing of BSB-2002 (day 0) * Ongoing treatment with chronic immunosuppressants (e.g., cyclosporine or systemic steroids at any dose) * Unstable cardiac disease as defined by one of the following: * Cardiac events such as myocardial infarction (MI) within the past 6 months * NYHA (New York Heart Association) heart failure class III-IV * Uncontrolled atrial fibrillation or hypertension * Uncontrolled bacterial, viral, or fungal infections at time of enrollment * Other active malignancy that requires treatment. Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * Females only: Pregnant or breastfeeding * Any other condition that would, in the Investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures or interference with study participation or data interpretation * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)