Testing the Addition of an Anti-cancer Drug, Hu5F9-G4 (Magrolimab), to the Usual Chemotherapy Treatment (Mogamulizumab) in T-Cell (a Type of Immune Cell) Lymphoma That Has Returned After Treatment or Does Not Respond to Treatment

Inclusion Criteria: * Diagnosis of either mycosis fungoides (MF) or Sezary syndrome (SS) by the World Health Organization (WHO) 2016 classification (Swerdlow et al., 2017), stage IB-IV by modified International Society for Cutaneous Lymphomas (ISCL)/ European Organization of Research and Treatment of Cancer (EORTC) classification (Olsen et al., 2011), without large cell transformation (LCT) at the time of screening. Patients with a history of prior LCT are permitted * Patients must have had at least one prior course of systemic therapy * Age \>= 18 years * Because no dosing or adverse event data are currently available on the use of Hu5F9-G4 (magrolimab) in combination with mogamulizumab in patients \<18 years of age, children are excluded from this study, but will be eligible for future pediatric trials * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Hemoglobin \>= 9.5 g/dL and transfusion independence (defined as not requiring more than 2 units of red blood cell \[RBC\] transfusions during the 4-week period prior to screening) * Absolute neutrophil count \>= 1,000/mcL * Platelets \>= 75,000/mcL * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) except for subjects with Gilbert's syndrome or genetic equivalent * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x institutional ULN * Glomerular filtration rate (GFR) \>= 40 mL/min/1.73 m\^2 by the Cockcroft-Gault formula * Patients must meet the following minimum wash-out window from previous treatments to the first treatment * \>= 3 weeks for systemic anti-cancer therapies * \>= 2 weeks for phototherapy, local radiation therapy, topical high potency corticosteroid, topical retinoid, topical nitrogen mustard, or topical toll-like receptor (TLR)-agonist * \>= 12 weeks for total skin electron beam therapy Participants with rapidly progressive malignant disease may be enrolled prior to completion of the above periods with approval of the protocol director * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression for a minimum of 3 months * 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 * The effects of Hu5F9-G4 (magrolimab) on the developing human fetus are unknown. For this reason and because monoclonal antibody agents as well as other therapeutic agents used in this trial (mogamulizumab) are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation and continue for 4 months after the last dose of both Hu5F9-G4 (magrolimab) and mogamulizumab. Effective contraception is defined as oral contraceptives, double barrier method (condom plus spermicide or diaphragm plus spermicide) or practice true abstinence from heterosexual intercourse. Women of childbearing potential includes any female who has experienced menarche and has not undergone successful surgical sterilization or is not postmenopausal (defined as amenorrhea \>= 12 consecutive months). Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol who are sexually active with women of childbearing potential and who have not had vasectomies must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of Hu5F9-G4 (magrolimab) administration * Female patients of childbearing potential must not be nursing or planning to be pregnant and must have a negative urine or serum pregnancy test within 30 days before randomization and within 72 hours before the first administration of study treatment * Female patients of childbearing potential must be willing to use one highly effective method of contraception during the study and continue for 4 months after the last dose of study treatment * Male patients who are sexually active with a woman of childbearing potential (WOCBP) and who have not had vasectomies must be willing to use a barrier method of contraception (condom plus spermicidal gel) and refrain from sperm donation during the study and for 4 months after the last dose of study treatment. If the partner is pregnant, male patients must use barrier method contraception (condom) during the study and for 4 months after the last dose of study treatment to prevent fetal exposure to study treatment * Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible * Willing to comply with clinic visit schedule and procedures including mandatory biopsies Exclusion Criteria: * Prior treatment with Hu5F9-G4 (magrolimab) or any agent targeting CD47-SIRPalpha * Prior progression of disease with mogamulizumab * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1) with the exception of alopecia and lymphopenia (any grade permitted). Residual peripheral neuropathy must have improved to grade 2 or better * Patients who are receiving any other investigational agents * Allogeneic hematopoietic stem cell transplant recipients with any graft-versus-host disease within the previous 3 months or requiring immunosuppression * Active autoimmune disease that has required systemic immunosuppressive medication within the previous 3 months * Active herpes simplex or herpes zoster. Subjects on prophylaxis for herpes who have no active signs of active infection, and whose last active infection was more than 6 months ago, may enter the study, and should continue to take the prescribed medication for the duration of the study * History of allergic reactions attributed to compounds of similar chemical or biologic composition to Hu5F9-G4 (magrolimab), other monoclonal antibodies, or other agents (mogamulizumab) used in study * Significant cardiopulmonary disease defined as * Acute myocardial infarction within the last 6 months * Unstable angina * Congestive heart failure New York Heart Association (NYHA) class III-IV * Patients with uncontrolled intercurrent illness requiring antibiotics. Patients on prophylactic antibiotics for non-complicated staphylococcus colonization/infection are eligible * Patients with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are excluded * Patients with psychiatric illness/social situations or substance abuse that would limit compliance with study requirements * Pregnant women are excluded from this study because Hu5F9-G4 (magrolimab) is monoclonal antibody agent with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with Hu5F9-G4 (magrolimab), breastfeeding should be discontinued if the mother is treated with Hu5F9-G4 (magrolimab). These potential risks may also apply to other agents used in this study (mogamulizumab) * Patients with RBC transfusion dependence, defined as requiring more than 2 units of RBCs transfused during the 4-week period prior to screening. RBC transfusions are permitted during the screening period and prior to enrollment * Patients with prior autoimmune thrombocytopenia, hemolytic anemia or Evans syndrome requiring treatment in the last 12 months * Patients on the following medications at the time of enrollment: * Immunotherapy or immunosuppressive drugs (e.g. chemotherapy or systemic corticosteroids) EXCEPT for the following: * Intranasal, inhaled, or local steroid injection (e.g. intra-articular injection) * Systemic corticosteroids at physiologic doses =\< 10 mg/day of prednisone or equivalent, if patient has been on a stable dose for at least 2 weeks prior to the first treatment * Low and medium potency topical corticosteroids are permitted if patient has been on a stable dose for at least 2 weeks prior to the first treatment * Steroids as premedication for hypersensitivity reactions (e.g. computed tomography \[CT\] scan premedication) * Growth factors (granulocyte colony stimulating factor or granulocyte macrophage colony stimulating factor) EXCEPT for erythropoietin and darbepoetin alpha * Herbal remedies with immunostimulating properties (e.g., mistletoe extract) or known to potentially interfere with major organ function (e.g. hypericin) * Live vaccines are not allowed while participating in the trial. Examples of live vaccines include, but are not limited to, the following: measles, mumps, rubella, chicken pox, yellow fever, rabies, Bacillus Calmette-Guerin (BCG), typhoid (oral) vaccine, and intranasal influenza vaccines (e.g., Flu-Mist). However, seasonal influenza vaccines for injection are generally killed virus vaccines and are allowed