Objective: Multi-channel speech enhancement (MCSE) focuses on recovering clean speech signals by leveraging both spatial relationships and temporal spectral information captured by microphone arrays. MCSE plays a crucial role in applications such as teleconferencing, human-computer interaction, and distant speech recognition, where enhancing speech quality and intelligibility is critical. However, existing approaches often struggle in challenging acoustic environments with low signal-to-noise ratios (SNRs) and high reverberation. While these methods excel at noise reduction and distortion minimization, they often fail to preserve the spatial and acoustic structure of speech in adverse environments. Specifically, methods that rely on networks to implicitly learn joint spectral and spatial information tend to struggle with accurately predicting beamforming weights, which are crucial for noise suppression. This challenge frequently results in speech distortion, such as loss of harmonic structures and decreased speech intelligibility. Methods: To address these issues, this study proposes a two-stage MCSE framework that explicitly incorporates the harmonic structure of speech. The first stage incorporates integrating a speech harmonic information extraction module into a UNet-based beamforming network; this module is designed to capture and retain the harmonic structure, which is fundamental to human auditory perception and speech clarity. The second stage introduces a residual iterative correction module to refine the speech signal; this stage focuses on addressing fiber acoustic structures that were not fully processed during the initial stage. By progressively reducing residual distortions, this approach ensures to improve speech quality, even in environments with low SNRs and significant reverberation, conditions where traditional methods typically fall short. Results: This study evaluated the proposed framework on a dataset derived from LibriSpeech, simulating various challenging acoustic environments with varying SNRs levels and reverberation conditions. The results demonstrated significant improvements over existing MCSE techniques in noise suppression and speech distortion reduction. Specifically, in low SNRs and highly reverberant environments, the beamforming network with harmonic structure information preserved the spatial and spectral characteristics of the speech signal better than traditional methods. Compared with approaches relying solely on implicitly learned spectral-spatial information, the proposed model showed greater effectiveness in retaining speech intelligibility and clarity. The inclusion of harmonic structure information allowed proposed framework to achieve better differentiation between speech and noise, producing more robust and reliable enhancement results under adverse conditions. The residual iterative correction stage further improved model performance by refining unprocessed acoustic structures, thus reducing residual distortions and enhancing spectral richness in the enhanced speech signal. Conclusions: The proposed two-stage MCSE framework successfully addresses the limitations of traditional MCSE methods by explicitly modeling and preserving the harmonic structure of speech. By integrating this information, the framework enhances spatial-spectral processing, enabling superior noise suppression and restoration of speech clarity, particularly in complex acoustic environments. The findings of this study indicate the significance of harmonic structures in human auditory perception and how their preservation contributes to improved intelligibility. The inclusion of the residual iterative correction stage in proposed framework ensures that finer acoustic details are addressed, allowing the model to perform reliably in low SNRs and high reverberation scenarios where conventional approaches often fail. This research underscores the importance of incorporating explicit acoustic structure information into MCSE systems, paving the way for more advanced speech enhancement models that can reliably address acoustic challenges in real-world. The research results of this paper can provide a reference for multi-channel speech enhancement schemes in extreme acoustic scenarios.