Inclusion Criteria:  Diagnosis:  Relapsed or refractory acute lymphoblastic leukemia or lymphoma with ≥1% bone marrow disease as measured by flow cytometry, PCR, or next generation sequencing. However, if an adequate bone marrow sample cannot be obtained, patients may be enrolled if there is unequivocal evidence of leukemia with ≥ 5% blasts in the peripheral blood. Patients with 1-4.99% bone marrow involvement must have disease confirmed in one of the following ways: an alternative minimal residual disease assay (e.g. flow cytometry and PCR or NGS), cytogenetic abnormality consistent with patient's leukemia, FISH abnormality, or a second bone marrow with MRD ≥1% separated by 1-4 weeks. Patients with ≥5% bone marrow disease by a single measurement as measured by flow cytometry, PCR, or next generation sequencing do not require a second confirmatory test. Refractory disease is defined as residual leukemia ≥1% after at least 2 prior lines of frontline therapy with curative intent. Patients in exploratory cohort I must have measurable extramedullary disease but may have <1% bone marrow disease. Patients in exploratory cohort M must have ≥1% bone marrow disease as measured by flow cytometry of mixed phenotype acute leukemia (MPAL)/ acute leukemia of ambiguous lineage (ALAL). Age ≥4 to < 30 years. Patients ≥ 22 years old are only eligible for exploratory cohort O. Sites may have different (lower) maximum ages based on institutional guidelines but may not exceed 30 years. Patient weighs ≥ 20 kg. Patient is able to swallow pills. Lansky/Karnofsky score is ≥ 60%. The Lansky performance score should be used for participants < 16 years and the Karnofsky performance score for participants ≥ 16 years.  Participant has adequate organ function as defined by the following:  Direct bilirubin ≤ 1.5x the institutional upper limit of normal (ULN). At institutions which do not obtain a direct bilirubin in patients with a normal total bilirubin, a normal total bilirubin may be used as evidence that the direct bilirubin is not > 1.5x the ULN. Aspartate transaminase (AST) and alanine transaminase (ALT) < 3.0 x the ULN unless increase is attributable to leukemic involvement. Normal creatinine for age or a calculated creatinine clearance ≥ 60 mL/min/1.73 m^2.  Left ventricular ejection fraction (LVEF) ≥ 40% or shortening fraction ≥ 25%.  Patients with a history of reduced LVEF which subsequently improved with medical management are eligible if they meet the criteria above. Patients must have fully recovered from the acute effects of all prior therapy (defined as resolution of all such toxicities to ≤ Grade 2). For patients with prior hematopoietic stem cell transplant (HSCT), at least 90 days must have elapsed since transplant, the patient cannot have evidence of active graft-versus-host disease (GVHD), and they must be off calcineurin inhibitors for ≥4 weeks, and off other immunosuppression for ≥2 weeks. Patients with Down Syndrome/ germline Trisomy 21 are eligible for Block 1 and Block 2b therapies but are ineligible for Block 2a therapy. Patients with Down Syndrome and CD19-negative disease are off therapy after the response evaluation to Block 1.  Prior therapy  ≥14 days must have elapsed since the completion of cytotoxic therapy, with the exception of standard maintenance therapy (glucocorticoids, vincristine, methotrexate, 6-mercaptopurine), tyrosine kinase inhibitors, and steroids. Cytoreduction with prednisone, methylprednisolone, or hydroxyurea for ≤ 120 hours (5 days) in patients with hyperleukocytosis or extramedullary disease compromising organ function can be initiated and continued until up to 24 hours prior to the start of protocol therapy. At least 21 days must have elapsed since completion of therapy with a biologic agent excluding blinatumomab. For agents that have known adverse events occurring beyond 21 days after administration, this period prior to enrollment must be extended beyond the time during which adverse events are known to occur. At least 7 days must have elapsed since blinatumomab infusion and patients must have recovered from all toxicities as described above. Intrathecal cytotoxic therapy: No waiting period is required for patients having received intrathecal cytarabine, methotrexate, and/or hydrocortisone. Intrathecal chemotherapy given at the time of diagnostic LP to evaluate for relapse prior to study enrollment is allowed. Patient has not had prior exposure to navitoclax Male or female participant of reproductive potential must agree to use appropriate methods of contraception for the duration of study treatment and for at least 30 days after last dose of protocol treatment.  Additional criteria for exploratory cohorts  Cohort I: Diagnosis of isolated extramedullary relapse as defined by bone marrow blasts of <1% AND 1) central nervous system white blood cell count (WBC) of ≥ 5WBC/mL with blasts or 2) biopsy confirmed extramedullary leukemia. Cohort M: Diagnosis of relapsed or refractory mixed phenotype acute leukemia (MPAL)/ acute leukemia of ambiguous lineage (ALAL). Cohort N: Patients with relapsed or refractory ALL who, in the view of the provider, are unable to tolerate further asparaginase therapy due to prior toxicities. Cohort O: Patients with relapsed or refractory ALL who are ages 22-29.9 years. This cohort may not enroll patients at all sites based on institutional guidelines or capacity.  Exclusion Criteria:  Known HIV infection or active hepatitis B (defined as hepatitis B surface antigen-positive) or C (defined as hepatitis C antibody-positive). Pregnant or lactating (female participant of childbearing potential must have negative serum or urine pregnancy test required within 7 days prior to start of treatment).  Concomitant medications and food:  Treatment with moderate or strong cytochrome P450 3A (CYP3A) inhibitors within 3 days of starting protocol therapy. Treatment with moderate or strong CYP3A inducers within 7 days of starting protocol therapy. Administration or consumption within 3 days prior to the first dose of study drug or grapefruit or grapefruit products, Seville oranges (including marmalade containing Seville oranges), or star fruit. Inability or unwillingness of research participant or legal guardian/representative to give written informed consent.