Nowadays, numerous online platforms can be described as multi-modal heterogeneous networks (MMHNs), such as Douban's movie networks and Amazon's product review networks. Accurately categorizing nodes within these networks is crucial for analyzing the corresponding entities, which requires effective representation learning on nodes. However, existing multi-modal fusion methods often adopt either early fusion strategies which may lose the unique characteristics of individual modalities, or late fusion approaches overlooking the cross-modal guidance in GNN-based information propagation. In this paper, we propose a novel model for node classification in MMHNs, named Heterogeneous Graph Neural Network with Inter-Modal Attention (HGNN-IMA). It learns node representations by capturing the mutual influence of multiple modalities during the information propagation process, within the framework of heterogeneous graph transformer. Specifically, a nested inter-modal attention mechanism is integrated into the inter-node attention to achieve adaptive multi-modal fusion, and modality alignment is also taken into account to encourage the propagation among nodes with consistent similarities across all modalities. Moreover, an attention loss is augmented to mitigate the impact of missing modalities. Extensive experiments validate the superiority of the model in the node classification task, providing an innovative view to handle multi-modal data, especially when accompanied with network structures. The full version including Appendix is available at http://arxiv.org/abs/2505.07895.