Trial Evaluating the Efficacy of CARBOPLATIN in Metastatic Prostate Cancer With Gene Alterations in the Homologous Recombination Pathway

Inclusion Criteria: * Patients \> 18 years old * Patients with adenocarcinoma or poorly differentiated prostate carcinoma, histologically confirmed (small-cell histology or high-grade neuroendocrine histology excluded) * Tumor presenting a somatic pathogenic variant likely to alter the homologous recombination pathway previously detected on a tumor biopsy or on circulating tumor DNA, or germinal mutation among the list of genes defined in the study * Castration-resistant tumor defined by progression despite well-conducted androgen deprivation treatment: testosterone ≤50ng /dL agonist / antagonist of luteinizing hormone-releasing hormone (LHRH) or surgical castration. The patient must agree to continue concomitant LHRH-mediated (agonist or antagonist) therapy throughout the duration of the study regimen for patients with no history of surgical castration. * Patients must have performed at least one line of chemotherapy by taxane in case of castration resistance: * Patients who have received docetaxel treatment in a hormone-sensitive situation must have received at least treatment with cabazitaxel in case of castration resistance * Patients who have not received chemotherapy in a hormone-sensitive situation must have received docetaxel AND cabazitaxel or have a contraindication to discontinue treatment. * Patients must have been treated with at least 2nd generation hormone therapy (eg, abiraterone acetate or enzalutamide) * Patients may have been treated with a poly (ADP-ribose) polymerase inhibitor (PARP) * Performance Status \<2 * Metastatic disease progressive Exclusion Criteria: * Absence of previous treatment with taxane in situation of sensitivity or resistance to castration. * Absence of previous treatment with cabazitaxel in case of resistance to castration (except contraindication explaining the non-administration of treatment) * No treatment with 2nd generation hormone therapy (eg abiraterone acetate or enzalutamide) unless contraindicated to explain non-administration of treatment * Previous treatment with platinum * Symptomatic and untreated central nervous system (CNS) metastases. Patients with asymptomatic and pre-treated CNS metastases are included if they are clinically stable (not requiring corticosteroid therapy for 28 days) and must have a brain MRI evaluation at screening and during follow-up.