PO Ixazomib in Combination With Chemotherapy for Childhood Relapsed or Refractory Acute Lymphoblastic Leukemia and Lymphoblastic Lymphoma

Inclusion Criteria: * Age Patients must be ≤21 years of age at the time of enrollment. 1. Phase 1 - Initial enrollment will be restricted to patients \< 18 years of age until 9 such patients are enrolled 2. Phase 2 - Initial enrollment will be restricted to patients \< 18 years of age until 6 such patients are enrolled (applies to Stratum A only) * Diagnosis Patients must have a diagnosis of relapsed/refractory ALL or LLy with or without extramedullary disease (including CNS2 and CNS3). Patient with mixed phenotype ALL or mature B (Burkitt-like) leukemia are not eligible. 1. Patients with ALL must have ≥ 5% blasts by morphology. 2. Patients with LLy must have measurable disease documented by clinical, radiologic or histologic criteria * Performance Level Karnofsky ≥ 50% for patients \> 16 years of age and Lansky ≥ 50% for patients ≤ 16 years of age. * Prior Therapy A. Prior therapeutic attempts * Phase 1 - Any patients with relapsed/refractory ALL or LLy * Phase 2 1. B-cell ALL/LLy: all patients must have failed two or more therapeutic attempts. 2. T-cell ALL/LLy: all patients must have failed one or more therapeutic attempts. B. Recent prior chemotherapy Patients must have fully recovered from the acute toxic effects of all prior chemotherapy, immunotherapy, or radiotherapy prior to entering this study. * Myelosuppressive chemotherapy: At least 14 days must have elapsed since the completion of myelosuppressive therapy. However, patients may receive any of the following medications within 14 days without a "wash-out" period: * Hydroxyurea: Hydroxyurea can be initiated and/or continued for up to 24 hours prior to the start of protocol therapy. * "Maintenance-style" therapy - Therapy including vincristine (dosed at a maximum of one-time weekly), oral 6-mercaptopurine, oral methotrexate (dosed at a maximum of one-time weekly), dexamethasone (dosed at ≤ 3 mg\*/m\^2/dose twice daily), and prednisone (dosed at ≤ 20 mg\*/m\^2/dose twice daily) can be initiated and/or continued for up to 24 hours prior to the start of protocol therapy. * Doses can be rounded to adjust for pill size C. Hematopoietic stem cell transplant: Patients who have experienced their relapse after a HSCT are eligible, provided they have no evidence of acute or chronic Graft-versus-Host Disease (GVHD), are not receiving GVHD prophylaxis or treatment, and are at least 90 days post-transplant at the time of enrollment. D. Hematopoietic growth factors: It must have been at least 7 days since the completion of therapy with G-CSF or other growth factors at the time of enrollment. It must have been at least 14 days since the completion of therapy with long-acting filgrastim (pegfilgrastim or biosimilar) E. Biologic (anti-neoplastic agent): At least 7 days since the last dose of a biologic agent. For agents that have known adverse events occurring beyond 7 days after administration, this period must be extended beyond the time during which adverse events are known to occur. The duration of this interval must be discussed with the study chair 1. Monoclonal antibodies: At least 3 half-lives of the antibody or 21 days (whichever is shorter) must have elapsed after the last dose of monoclonal antibody. (i.e., blinatumomab half-life = 6 hours, therefore washout is 18 hours; inotuzumab half-life = 37 days therefore washout is 21 days; rituximab half-life = 66 days, therefore washout is 21 days). If steroids are being used to modify immune-related adverse events of antibody therapy, at least 14 days must have elapsed since the last dose of corticosteroid. 2. Immunotherapy: At least 30 days after the completion of any type of immunotherapy, e.g., tumor vaccines, CAR T cells. If steroids are being used to modify immune-related adverse events of immunotherapy, at least 14 days must have elapsed since the last dose of corticosteroid. F. XRT: Craniospinal XRT is prohibited during protocol therapy. No washout period is necessary for radiation given to any extramedullary site other than CNS; ≥90 days must have elapsed if prior total body irradiation (TBI) or craniospinal XRT. G. Anthracyclines: Patients must have had a lifetime exposure of \<400 mg/m\^2 of doxorubicin equivalents of anthracyclines. H. Proteasome inhibitors: Patients with a prior exposure to proteasome inhibitors (e.g., bortezomib, carfilzomib) are eligible as long as the patient demonstrated at least a partial response to a proteasome inhibitor with chemotherapy combination. This criteria only applies to the Phase 2 portion of the study. -Renal and hepatic function Patients must have adequate renal and hepatic functions as indicated by the following laboratory values: A. Adequate renal function defined as: Patient must have a calculated creatinine clearance or radioisotope GFR ≥ 70ml/min/1.73m\^2 OR a normal serum creatinine based on age/gender B. Adequate Liver Function Defined as: Direct bilirubin ≤ 1.5 x upper limit of normal (ULN) for age or normal (except in the presence of Gilbert's syndrome), AND alanine transaminase (ALT) ≤ 5 x ULN for age. The hepatic requirements are waived for patients with known or suspected liver involvement by leukemia or lymphoma. This must be reviewed by and approved by the study chair or vice chair. * Adequate Cardiac Function Defined as: Shortening fraction of ≥ 27% by echocardiogram, OR ejection fraction of ≥ 50% by radionuclide angiogram (MUGA). * Reproductive Function A. Female patients of childbearing potential must have a negative urine or serum pregnancy test confirmed within 2 weeks prior to enrollment. B. Female patients with infants must agree not to breastfeed their infants while on this study. C. Male and female patients of child-bearing potential must agree to use an effective method of contraception approved by the investigator during the study and for a minimum of 6 months after study treatment. * Informed Consent Patients and/or their parents or legal guardians must be capable of understanding the investigational nature, potential risks and benefits of the study. All patients and/or their parents or legal guardians must sign a written informed consent. Age appropriate assent will be obtained per institutional guidelines. To allow non-English speaking patients to participate in this study, bilingual health services will be provided in the appropriate language when feasible. * All institutional, FDA, and OHRP requirements for human studies must be met. Exclusion Criteria: Patients will be excluded if they have isolated CNS or testicular disease. Patients will be excluded if they have ≥ grade 2 peripheral sensory or motor neuropathy (defined by the Modified "Balis" Pediatric Scale of Pediatric Neuropathies) at the time of enrollment. Patients will be excluded if they have a known allergy or intolerance to any of the drugs used in the study - except for Pegaspargase for which asparaginase Erwinia chrysanthemi (recombinant)-rywn (Rylaze®) or (if available) crisantaspase (Erwinase®), may be substituted for allergy to Pegaspargase Patients will be excluded if they have a systemic fungal, bacterial, viral or other infection that is exhibiting ongoing signs/symptoms related to the infection without improvement despite appropriate antibiotics or other treatment. The patient needs to be off pressors and have negative blood cultures for 48 hours. Patients will be excluded if there is a plan to administer non-protocol chemotherapy, radiation therapy, or immunotherapy during the study period. Patients will be excluded if they have significant concurrent disease, illness, psychiatric disorder or social issue that would compromise patient safety or compliance with the protocol treatment or procedures, interfere with consent, study participation, follow up, or interpretation of study results. Patients with DNA fragility syndromes (such as Fanconi anemia, Bloom syndrome) are excluded. Patients will be excluded if they have had a lifetime exposure of ≥400 mg/m2 doxorubicin equivalents of anthracyclines (anthracycline equivalence to doxorubicin conversion see appendix iv) . Concomitant medications Investigational drugs: Patients currently receiving another investigational drug are not eligible. Anti-GVHD agents post transplant: patients who are receiving cyclosporine, tacrolimus or other agents to prevent graft-versus-host disease post hematopoetic stem cell transplant are not eligible. CYP3A4 agents: patients who are currently receiving drugs that are strong inducers of CYP3A4 are not eligible. Patients with Ph+ALL and Ph-like ALL who are currently receiving TKI therapy