Reducing power consumption subject to quality-of-service (QoS) provisions is a critical task for next-generation networking. Recent research in resource management for orthogonal frequency-division multiple-access (OFDMA) systems has generally assumed the availability of perfect channel state information at the transmitter (CSIT) or imperfect CSIT with small uncertainty. Nevertheless, such approaches deliver resource scheduling strategies with high transmitting power because, in real environments, large channel feedback delays and estimation errors cause high CSIT imperfectness. Furthermore, most existing works treat various QoS requirements from mobile users as homogenous, although they are heterogeneous in nature. In this paper, we address these issues by proposing a new power-efficient adaptive error-tolerant cross-layer scheduling scheme for OFDMA systems (PE-AETS). Our target is to minimize the transmitting power by considering heterogeneous QoS requirements and data outage due to imperfect CSIT. The proposed scheme adopts a robust power-bit loading (PBL) method that adjusts power and data rates across subcarriers with increased system resilience to channel errors. We develop a statistical queuing model to express the delay limitation of each user with an equivalent cross-layer constraint, and we apply subcarrier time-sharing relaxation to formulate a convex optimization problem. Finally, we utilize Lagrangian optimization to propose a joint power and subcarrier allocation policy with guaranteed convergence to optimal solutions and linear complexity. Various simulation scenarios confirm the superior performance of the proposed PE-AETS over relevant cross-layer approaches.