This research focuses on the predictive energy management of connected human-driven hybrid electric vehicles (HEVs) to improve their fuel efficiency while robustly satisfying system constraints. We propose a hierarchical control framework that effectively exploits long-term and short-term decision-making benefits by integrating real-time traffic data into the energy management strategy. A pseudo-spectral optimal controller (PSOC) with discounted cost is utilized at the high level to find an approximate optimal solution for the entire driving cycle. At the low-level, a long short-term memory neural network (NN) is developed for higher quality velocity predictions over the low-level's short time horizons. Tube-based model predictive controller is then used at the low level to ensure constraints satisfaction in the presence of velocity prediction errors. Simulation results over real-world traffic data show an improvement in fuel economy for the proposed controller that is real-time applicable and robust to the driving cycle's uncertainty.