Analyzing the basal ganglia following an early brain lesion is crucial due to their noteworthy role in sensory–motor functions. However, the segmentation of these subcortical structures on MRI is challenging in children and is further complicated by the presence of a lesion. Although current deep neural networks (DNN) perform well in segmenting subcortical brain structures in healthy brains, they lack robustness when faced with lesion variability, leading to structural inconsistencies. Given the established spatial organization of the basal ganglia, we propose enhancing the DNN-based segmentation through post-processing with a graph neural network (GNN). The GNN conducts node classification on graphs encoding both class probabilities and spatial information regarding the regions segmented by the DNN. In this study, we focus on neonatal arterial ischemic stroke (NAIS) in children. The approach is evaluated on both healthy children and children after NAIS using three DNN backbones: U-Net, UNETr, and MSGSE-Net. The results show an improvement in segmentation performance, with an increase in the median Dice score by up to 4% and a reduction in the median Hausdorff distance (HD) by up to 93% for healthy children (from 36.45 to 2.57) and up to 91% for children suffering from NAIS (from 40.64 to 3.50). The performance of the method is compared with atlas-based methods. Severe cases of neonatal stroke result in a decline in performance in the injured hemisphere, without negatively affecting the segmentation of the contra-injured hemisphere. Furthermore, the approach demonstrates resilience to small training datasets, a widespread challenge in the medical field, particularly in pediatrics and for rare pathologies.