Fostamatinib for the Treatment of Lower-risk Myelodysplastic Syndromes or Chronic Myelomonocytic Leukemia Who Have Failed Therapy With Hypomethylating Agents

Inclusion Criteria: * Age \>= 18 years as MDS is a very rare disease in the pediatric setting * Diagnosis of MDS or CMML according to World Health Organization (WHO) and very low, low or intermediate risk by Revised International Prognostic Scoring System (IPSS-R) (with IPSS-R score of =\< 3.5) * Patients need to have not responded to prior therapy with hypomethylating agents (HMAs). These could include azacitidine, decitabine, SGI-110, ASTX727. Patients will need to have received at least 4 cycles of HMA. Patients with relapse or progression after any number of cycles of HMA by IWG 2006 criteria will also be candidates. Patients with MDS with isolated del(5q) must have received prior therapy with lenalidomide * Cytopenias, in the form of anemia and thrombocytopenia, as follows: * Hemoglobin \< 10 g/dL OR * Platelets \< 75 x10\^9/L OR * Transfusion dependency, defined as the receipt of any red blood cell or platelet transfusions within at least 28 days prior to the start of study treatment * Patient (or patient's legally authorized representative) must have signed an informed consent document indicating that the patient understands the purpose of and procedures required for the study and is willing to participate in the study * Total bilirubin \< 2 x upper limit of normal (ULN) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) \< 1.5 x ULN * Serum creatinine clearance \> 30 mL/min and no end/stage renal disease (using Cockcroft-Gault) * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Hydroxyurea for control of leukocytosis is allowed at any time prior to or during study if considered to be in the best interest of the patient * Both females of childbearing potential and males with female partners of childbearing potential must agree to use contraception during the study period and for at least one month after the last dose Exclusion Criteria: * No prior therapy for MDS or CMML * Uncontrolled infection not adequately responding to appropriate antibiotics * Absolute neutrophil count (ANC) \< 0.5 x 10\^9 k/uL * Uncontrolled or poorly controlled hypertension, defined as systolic blood pressure \>= 135 mmHg or diastolic blood pressure \>= 85 mmHg, whether or not the subject is receiving anti-hypertensive treatment * Female patients who are pregnant or lactating * Patients with reproductive potential who are unwilling to following contraception requirements (including condom use for males with sexual partners, and for females: prescription oral contraceptives \[birth control pills\], contraceptive injections, intrauterine devices \[IUD\], double-barrier method \[spermicidal jelly or foam with condoms or diaphragm\], contraceptive patch, or surgical sterilization) throughout the study. Reproductive potential is defined as no previous surgical sterilization or females that are not post-menopausal for 12 months * Female patients with reproductive potential who do not have a negative urine or blood beta-human chorionic gonadotropin (beta human chorionic gonadotropin \[HCG\]) pregnancy test at screening * History of an active malignancy within the past 2 years prior to study entry, with the exception of: a. Adequately treated in situ carcinoma of the cervix uteri b. Adequately treated basal cell carcinoma or localized squamous cell carcinoma of the skin or any other malignancy with a life expectancy of more than 2 years * Patients receiving any other concurrent investigational agent or chemotherapy, radiotherapy, or immunotherapy (within 14 days of initiating study treatment) * Evidence of graft versus host disease or prior allo-stem cell transplantation within 6 months of Cycle 1 Day 1 or receiving immunosuppressants following a stem-cell procedure