This paper investigates the age of information (AoI) for a radio frequency (RF) energy harvesting (EH) enabled network, where a sensor first scavenges energy from a wireless power station and then transmits the collected status update to a sink node. To capture the thirst for the fresh update becoming more and more urgent as time elapsing, urgency-aware AoI (U-AoI) is defined, which increases exponentially with the increment of time between two received updates. Due to EH, a waiting time is required at the sensor before transmitting the status update. An optimization problem is formulated to minimize the long-term average U-AoI under constraint of energy causality. A two-layer algorithm is presented to solve it, where the outer loop is designed based on Dinklebach's method, and the inner loop presents a semi-closed-form expression of the optimal waiting time policy based on Karush-Kuhn-Tucker (KKT) optimality conditions. Numerical results show that our proposed optimal transmission policy outperforms the zero time waiting policy and equal time waiting policy in terms of long-term average U-AoI, especially when the networks are in slight load. It also shows that the system U-AoI first decreases and then keeps unchanged with the increments of EH circuit's saturation level.