This letter studies a hybrid non-orthogonal multiple access (NOMA)-based wireless system, in which the users harvest energy from the received downlink signals in order to transmit on the uplink. We argue that the use of energy harvesting helps in maintaining distinct power levels when the users commence uplink NOMA transmissions. To evaluate the performance of the proposed model, we derive closed-form mathematical expressions of the average ergodic rate of the system. The results illustrate that: 1) this scheme is capable of establishing NOMA-based uplink communication solely on the basis of harvested energy and 2) power-splitting parameter shows positive correlation on the uplink rate performance, whereas the choice of time-splitting parameter favors either uplink or downlink data rate.