For a two-hop IoT system consisting of multiple energy harvesting sensors, a cache-enabled edge node, and multiple monitors, the status update control at the edge node, which has partial battery state information (pBSI) of the sensors, is formulated as a pBSI problem. The concept of inferred pBSI is introduced to reduce the noiseless single-sensor pBSI problem to a Markov decision process with a moderate state-space size, enabling the optimal policy to be obtained through a value iteration algorithm. A lower bound on the expected time-average on-demand age of information performance is established for the general single-sensor status update problem. For the single-sensor pBSI problem, a semi-closed-form policy called the current-next (CN) policy is proposed, along with an efficient post-update value iteration algorithm with a per-iteration time complexity proportional to the square of the battery capacity. A weighted-update-gain-competition (WUGC) approach is further leveraged to extend the CN policy to the multi-sensor case. Numerical results in the single-sensor case demonstrate the near-optimal performance of the CN policy across various energy arrival processes. Simulations for an IoT system with $100$ sensors reveal that the WUGC-CN policy outperforms the maximum-age-first policy and the random-scheduling-based CN policy under Bernoulli energy arrival processes.