Elastic storage has been reported to help flying insects save inertial power when flapping their wings. This motivates recent research and development of elastic storage for flapping-wing micro air vehicles (fwMAVs) and their ground (tethered) flight tests. The previous designs of spring-loaded transmissions are relatively heavy or bulky; they have not yet been adopted by freely hovering prototypes of fwMAVs, especially those with four flapping wings. It is not clear if partial elastic storage can still help save power for flapping flight while not overloading the motorized transmission. Here, we developed ultralight and compact film hinges as elastic storage for four flapping wings. This springassisted transmission was motor driven such that the wing beat frequency was higher than the natural frequency of elastically hinged wings. Our experiments show that spring recoil helps accelerate wing closing thus generating more thrust. When powered by a 3.18 g brushless motor, this 13.4 g fwMAV prototype with spring-assisted transmission can take off by beating four flexible wings (of 240mm span) with up to 21-22 g thrust generation at 22-23 Hz. Due to lower disk loading and high-speed reduction, indirect drive of the four elastically hinged wings can produce a thrust per unit of electrical power of up to 4.6 g/W. This electrical-power-specific thrust is comparable to that generated by direct drive of a propeller, which was recommended by the motor (AP-03 7000kv) manufacturer.