Recent years have witnessed the success of heterogeneous graph neural networks (HGNNs) in modeling heterogeneous information networks (HINs). In this paper, we focus on the benchmark task of HGNNs, i.e., node classification, and empirically find that typical HGNNs are not good at predicting the label of a test node whose receptive field (1) has few training nodes from the same category or (2) has multiple training nodes from different categories. A possible explanation is that their message passing mechanisms may involve noises from different categories, and cannot fully explore task-specific knowledge such as the label dependency between distant nodes. Therefore, instead of introducing a new HGNN model, we propose a general post-training framework that can be applied on any pretrained HGNNs to further inject task-specific knowledge and enhance their prediction performance. Specifically, we first design an auxiliary system that estimates node labels based on (1) a global inference module of multi-channel label propagation and (2) a local inference module of network schema-aware prediction. The mechanism of our auxiliary system can complement the pretrained HGNNs by providing extra task-specific knowledge. During the post-training process, we will strengthen both system-level and module-level consistencies to encourage the cooperation between a pretrained HGNN and our auxiliary system. In this way, both systems can learn from each other for better performance. In experiments, we apply our framework to four typical HGNNs. Experimental results on three benchmark datasets show that compared with pretrained HGNNs, our post-training framework can enhance Micro-F1 by a relative improvement of 3.9% on average. Code, data and appendix are available at https://github.com/GXM1141/HGPF.