Inclusion criteria applicable to all treatment arms:
* Histologically proven mature B-NHL (Diffuse Large B-Cell Lymphoma (DLBCL), Burkitt Lymphoma/Leukaemia or atypical Burkitt/Burkitt-like lymphoma, primary mediastinal large B-cell lymphoma (PMLBL), and mature B-NHL/Not Otherwise Specified (NOS)) at initial diagnosis
* Radiologically and/or histologically proven B-NHL in first relapse (only one prior line of therapy) or subsequent relapse (more than one prior line of therapy) or refractory(\*) B-NHL. (Note: relapses following prior targeted therapy must have continuing target positivity, confirmed by an established method).
* If relapse occurs more than two years after previous therapy, a biopsy must be performed
* Evaluable disease as per the international paediatric non-Hodgkin Lymphoma response criteria, including:
* at least one bi-dimensionally measurable nodal lesion \>1.5 cm in its longest dimension;
* or at least one bi-dimensionally measurable extra-nodal lesion \>1.0 cm in its longest dimension on computerised tomography (CT) or Magnetic Resonance Imaging (MRI);
* or bone marrow involvement (≥25% involvement from bone marrow, if only site of disease. Any standard method of assessment is acceptable i.e. cytomorphology, flow cytometry and/or immunohistochemistry);
* or, dependent on treatment arm, evaluable Central Nervous System (CNS) only disease (evaluable by imaging or Cerebrospinal Fluid (CSF) analysis)(\*\*)
* Age from birth to ≤25 years old at the time of trial entry
* Performance status ≥50 using Karnofsky or Lansky performance scores
* Life expectancy of ≥8 weeks
* Adequate bone marrow function documented by:
* Platelet count ≥50x 10\^9/L (no platelet transfusion therapy within seven days prior to treatment) unless bone marrow involvement(\*\*\*)
* Absolute neutrophil count (ANC) ≥0.75 x 10\^9/L (no granulocyte colony stimulating factor within 2 days prior to treatment) unless bone marrow involvement(\*\*\*)
* Adequate hepatic function documented by:
* Aspartate aminotransferase (AST) and/or alanine aminotransferase (ALT) ≤5 x upper limit of normal (ULN)
* Total bilirubin ≤1.5 X ULN \*\*\*Patients with known Gilbert syndrome will be excluded if the total bilirubin value is \>4 x ULN for the local general population
* Documented negative pregnancy test for female patients of childbearing potential within seven days prior to trial entry
* Patients of reproductive potential agrees to use effective contraception whilst on trial treatment and for 12 months following treatment discontinuation
* Written informed consent given by patient and/or parents/legal representative
Inclusion criteria applicable to treatment arm I only:
* Male patients of reproductive potential must agree not to donate sperm whilst on trial treatment and for 6 months following treatment discontinuation
* Adequate renal function, creatinine clearance \>45 ml/min by measurement or estimation (if creatinine levels are normal for the patient's age, using the Cockroft-Gault Equation is sufficient)
* For patients with bone marrow involvement(\*\*\*) or splenic sequestration, adequate bone marrow function documented by:
* Platelet count ≥25 x 10\^9/L (no platelet transfusion therapy within three days prior to treatment)
* Haemoglobin level ≥7 g/dL
* Absolute neutrophil count (ANC) ≥0.5 x 10\^9/L (no granulocyte colony stimulating factor within two days prior to treatment)
* Patients who have received CAR T-cell therapy or other cellular therapies more than 28 days prior must demonstrate recovery from acute toxicities and have measurable disease
Inclusion criteria applicable to treatment arm II only:
* Adequate renal function, by measured glomerular filtration rate (GFR) \>60 ml/min/1.73m\^2 (estimated GFR is not sufficient)
* For patients with bone marrow involvement(\*\*\*) or splenic sequestration, requirements for bone marrow function do not apply
(\*) Refractory disease
The following patients are considered to have refractory disease and can be included in this trial:
* Patients with who do not achieve PR or CR with last therapy
* Patients with partial response to last therapy (biopsy proven), with no evidence of progression
(\*\*) CNS only disease Patients with CNS only disease may be eligible depending on the treatment arm. Please refer to the relevant treatment arm specific eligibility criteria.
(\*\*\*) Bone marrow involvement Patients who have ≥ 25% blasts in the bone marrow are considered to have bone marrow involvement. For these patients, requirements for bone marrow function are dependent on treatment arm. Please refer to the relevant treatment arm specific eligibility criteria.
Exclusion Criteria:
* B-cell Acute Lymphoblastic Leukaemia (B-ALL)/B-cell Lymphoblastic Lymphoma (B-LBL)
* Patients within:
* 90 days after an allogenic HSCT procedure
* 45 days after an autologous HSCT procedure
* 28 days of experiencing graft versus host disease (GvHD) requiring systemic therapy, and/or immunosuppressive treatment
* 14 days of previous investigational treatment
* 28 days of receiving craniospinal radiation; or 14 days of any other radiation
* For patients who have received any CAR T-cell therapy or other cellular therapies, see treatment arm specific eligibility criteria
* Patients who have ongoing acute toxicities from most recent lymphoma directed therapy
* Patients with known DNA repair disorder or known primary immunodeficiency
* Patients who are pregnant or breastfeeding (exclusively or partially)
* Patients who cannot regularly be followed up in accordance with the protocol due to psychological, social, geographical or other issues
* Patients for whom non-compliance with treatment or trial procedures is expected
* Uncontrolled concomitant infection. Severe infection (such as sepsis, pneumonia, etc.) should be clinically controlled at the time of trial entry
* Known HIV positivity
* Hepatitis B carrier status, history of Hepatitis B Virus or positive serology. A patient is considered as a Hepatitis B Virus carrier or to have (had) Hepatitis B Virus infection in case of:
* Unimmunized and HBsAg and/or anti-HBs antibody and/or anti- HBc antibody positive,
* Immunized and HBsAg and/or anti-HBc antibody positive.
* Live vaccine within 28 days prior to trial entry
* Known history of hypersensitivity to any of the treatments or excipients
Exclusion criteria applicable to treatment arm I only:
* Central Nervous System (CNS) only disease
* Patients within 28 days of any CAR-T cell therapy or other cellular therapies
* Left ventricular shortening fraction (LVSF) \<27% or left ventricular ejection fraction (LVEF) \<50%, as determined by ECHO or MUGA, any evidence of pericardial effusion (except trace or physiological) as determined by an ECHO, and any clinically significant arrhythmias
* Known CD20 negative disease at initial diagnosis
* Seizure within the last 12 months
* Prior treatment with CD20 x CD3 bispecific therapy
* Known hypersensitivity to both allopurinol and rasburicase
Exclusion criteria applicable to treatment arm II only:
* Patients within 42 days of any CAR-T cell therapy or other cellular therapies
* Clinically significant (Grade ≥2) third space fluid accumulation (i.e., ascites requiring drainage or pleural effusion that is either requiring drainage or associated with shortness of breath)
* Steroid treatment for more than a total of seven days in the 14 days prior to trial entry
