Acute lymphoblastic leukemia (ALL) is the most common childhood cancer, with survivors frequently experiencing long-term neurocognitive morbidities. Here, we utilize the TOTXVI clinical trial data to elucidate the mechanisms underlying treatment-related neurocognitive side effects in pediatric ALL patients by incorporating brain connectivity network data. To enable such analysis, we propose a high-dimensional mediation analysis method with a novel network mediation structural shrinkage (NMSS) prior, which is particularly suited for analyzing high-dimensional brain structural connectivity network data that serve as mediators. Our method is capable of addressing the structural dependencies of brain connectivity networks including sparsity, effective degrees of nodes, and modularity, yielding accurate estimates of the high-dimensional coefficients and mediation effects. We demonstrate the effectiveness and superiority of the proposed NMSS method through simulation studies and apply it to the TOTXVI data, revealing significant mediation effects of brain connectivity on visual processing speed directed by IT intensity. The findings shed light on the potential of targeted interventions to mitigate neurocognitive deficits in pediatric ALL survivors.