Accurate trajectory prediction is crucial for safe and efficient autonomous driving, but handling partial observations presents significant challenges. To address this, we propose a novel trajectory prediction framework called Partial Observations Prediction (POP) for congested urban road scenarios. The framework consists of two stages: self-supervised learning (SSL) and feature distillation. In SSL, a reconstruction branch reconstructs the hidden history of partial observations using a mask procedure and reconstruction head. The feature distillation stage transfers knowledge from a fully observed teacher model to a partially observed student model, improving prediction accuracy. POP achieves comparable results to top-performing methods in open-loop experiments and outperforms the baseline method in closed-loop simulations, including safety metrics. Qualitative results illustrate the superiority of POP in providing reasonable and safe trajectory predictions.