Hybrid non-orthogonal multiple access (H-NOMA) is inherently an enabler of massive machine type communications, a key use case for sixth-generation (6G) systems. Together with backscatter communication (BackCom), it seamlessly integrates with the traditional orthogonal multiple access (OMA) techniques to yield superior performance gains. In this paper, we study BackCom assisted H-NOMA uplink transmission with the aim of minimizing power with imperfect channel state information (CSI), where a generalized representation for channel estimation error models is used. The considered power minimization problem with aggregate data constraints is both non-convex and intractable. For the considered imperfect CSI models, we use Lagrange duality and the majorization-minimization principle to produce a conservative approximation of the original problem. The conservative formulation is relaxed by incorporating slack variables and a penalized objective. We solve the penalized tractable approximation using a provably convergent algorithm with polynomial complexity. Our results highlight that, despite being conservative, the proposed solution results in a similar power consumption as for the nominal power minimization problem without channel uncertainties. Additionally, robust H-NOMA is shown to almost always yield more power efficiency than the OMA case. Moreover, the robustness of the proposed solution is manifested by a high probability of feasibility of the robust design compared to the OMA and the nominal one.