This paper addresses the critical challenge of mitigating hardware impairments (HWIs) in downlink cell-free massive MIMO (CF-mMIMO) networks while ensuring computational scalability. We propose a novel sequential hardware-aware (HW-aware) precoding technique that leverages the serial fronthaul topology to perform over-the-air HWI cancellation. This approach involves sequentially exchanging approximated user-perceived distortion information among successive access points (APs) for over-the-air HWI mitigation. Each AP independently computes its spatial multiplexing weights and transmits signals that counteract the distortions introduced by the preceding AP. We develop a problem formulation and present a closed-form solution for this method. For performance evaluation, we study two reference methods taken either from centralized massive MIMO literature (Tone Reservation [TR]) or tailored for CF-mMIMO networks (PAPR-aware precoding), both focusing on reducing the PAPR of OFDM signals in the downlink. Results indicate that the sequential HW-aware approach achieves a substantial increase in spectral efficiency (SE) in high-distortion scenarios, with an average SE increase factor of 1.8 under severe distortions. Additionally, the proposed method, which is executed locally, demonstrates better scalability, achieving a reduction of up to 40\% and 72\% in the total number of complex multiplications compared to the PAPR-aware and TR approaches, respectively. Finally, the sequential HW-aware precoder offers high performance even when applied to cost-effective APs with few antennas, presenting a promising and practical solution for HWI compensation in CF-mMIMO systems with serial fronthaul.