Dual mTorc Inhibition in advanCed/Recurrent Epithelial Ovarian, Fallopian Tube or Primary Peritoneal Cancer (of Clear Cell, Endometrioid and High Grade Serous Type, and Carcinosarcoma)

Inclusion Criteria: * Signed and dated written informed consent prior to admission to the study and initiation of any study procedures in accordance with ICH-GCP guidelines and to the local legislation * Females ≥ 18 years of age * Pathological diagnosis of ovarian, fallopian tube or primary peritoneal cancer, of clear cell, endometrioid or high grade serous subtype or carcinosarcoma. Local tumour board/MDT histological review is required and in mixed tumours more than 50% endometrioid, clear cell or high grade serous elements are required to define the predominant histology * Platinum-resistant (recurrence within 6 months of platinum treatment), or platinum refractory disease (recurrence during platinum treatment), patients having received at least one prior line of chemotherapy. Carboplatin and weekly paclitaxel are permitted as first line therapy and/or as therapy for recurrent platinum sensitive disease i.e. prior to platinum resistant relapse * Measurable disease as per Response Evaluation Criteria in Solid Tumours (RECIST) version 1.1 by CT or MRI * Fresh tumour biopsy during screening is compulsory if judged technically feasible by radiologist, unless the local site is unable to collect the sample due to COVID-19 capacity restrictions * Patients with a history of brain metastasis are eligible as long as all the following criteria are met: brain metastases must have been treated, have no evidence of progression or haemorrhage after treatment, have been off dexamethasone for 4 weeks prior to first study treatment, and no ongoing requirement for dexamethasone or anti-epileptic drugs * Available blocks for immunohistochemistry (IHC) and tissue microarray (TMA) or, if no block is available, 20 ordinary unstained slides (5µm sections) will be acceptable * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0-2 * Adequate organ and bone marrow function * Female patients who are postmenopausal for ≥ 1 year before the screening visit OR are surgically sterile OR if of childbearing potential, patient agrees to practice 1 highly effective and 1 additional effective method of contraception or true abstinence from informed consent to 90 days after the last dose of study treatment * For women of child-bearing potential, negative blood serum pregnancy test within 14 days prior to the first study treatment * Able to swallow oral medication Exclusion Criteria: * Previous treatment with PI3K, AKT, dual PI3K/mTOR inhibitors, mTORC1/2 inhibitors or mTORC1 inhibitors * Prior weekly single agent paclitaxel * Known allergy to paclitaxel and/or any excipients of investigational medicinal products that, in the investigator's opinion, precludes study treatment on clinical and/or safety grounds * Treatment with strong inhibitor/s and/or inducer/s of cytochrome P450 (CYP) 3A4 or CYP2C8 within 7 days of study treatment * Central nervous system (CNS) metastasis, for patients who have brain metastases, they will be eligible if their brain metastases must have been treated, have no evidence of progression or haemorrhage after treatment, have been off dexamethasone for 4 weeks prior to first study drug administration, and no ongoing requirement for dexamethasone or anti-epileptic drugs * Other clinically significant co-morbidities, such as uncontrolled pulmonary disease, active central nervous system disease, active infection, or any other condition that could compromise the patient's participation in the study * Known human immunodeficiency virus infection * Known hepatitis B surface antigen-positive, or known or suspected active hepatitis C infection * Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol * German sites only: Unable to be regularly followed up for any reason (geographic, familiar, social, psychological, housed in an institution e.g. prison because of a court agreement or administrative order) * German sites only: Subjects that are dependent on the sponsor (and/or contracted body e.g. CRO) or investigational site as well as on the investigator * Diagnosed or treated for another malignancy within 2 years before administration of the first dose of study treatment, or previously diagnosed with another malignancy and evidence of residual disease. Patients with nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection * Breast feeding or pregnant * Manifestations of malabsorption due to prior gastrointestinal (GI) surgery, GI disease, or for an unknown reason that may alter the absorption of TAK228. In addition, patients with enteric small bowel stomata are also excluded * Treatment with any investigational products, chemotherapy or radiotherapy within 28 days, or major surgery within 21 days of study treatment * History of any of the following within the last 6 months before administration of the first dose of study treatment: 1. Ischemic myocardial event, including angina requiring therapy and artery revascularisation procedures 2. Ischemic cerebrovascular event, including transient ischemic attack and artery revascularisation procedures 3. Requirement for inotropic support (excluding digoxin) or serious (uncontrolled) cardiac arrhythmia (including atrial flutter/fibrillation, ventricular fibrillation or ventricular tachycardia) 4. Placement of a pacemaker for control of rhythm 5. New York Heart Association (NYHA) Class III or IV heart failure 6. Pulmonary embolism * Significant active cardiovascular or pulmonary disease including: 1. Uncontrolled hypertension (i.e., systolic blood pressure \> 180 mm Hg, diastolic blood pressure \> 95 mm Hg). Use of anti-hypertensive agents to control hypertension before first dose of study treatment is allowed. 2. Pulmonary hypertension 3. Uncontrolled asthma or O2 saturation \< 90% by arterial blood gas analysis or pulse oximetry on room air 4. Significant valvular disease; severe regurgitation or stenosis by imaging independent of symptom control with medical intervention, or history of valve replacement 5. Medically significant (symptomatic) bradycardia 6. History of arrhythmia requiring an implantable cardiac defibrillator 7. Baseline prolongation of the rate-corrected QT interval (QTc) e.g. repeated demonstration of QTc interval \> 480 milliseconds, or history of congenital long QT syndrome, or additional risk factors for torsades de pointes (e.g. hypokalaemia, family history of long QT syndrome) and patients who use concomitant medications that prolong the QT/QTc interval * Patients receiving systemic corticosteroids (either IV or oral steroids, excluding inhalers or low-dose hormone replacement therapy) within 1 week before administration of the first dose of study treatment * Daily or chronic use of a proton pump inhibitor (PPI) and/or having taken a PPI within 7 days before receiving the first dose of study treatment * Poorly controlled diabetes mellitus defined as glycosylated haemoglobin (HbA1c) \> 7%; patients with a history of transient glucose intolerance due to corticosteroid administration may be enrolled in this study if all other inclusion/exclusion criteria are met