User-item interaction data in collaborative filtering and graph modeling tasks often exhibit power-law characteristics, which suggest the suitability of hyperbolic space modeling. Hyperbolic Graph Convolution Neural Networks (HGCNs) are a novel technique that leverages the advantages of GCN and hyperbolic space, and then achieves remarkable results. However, existing HGCN methods have several drawbacks: they fail to fully leverage hyperbolic space properties due to arbitrary embedding initialization and imprecise tangent space aggregation; they overlook auxiliary information that could enrich the collaborative graph; and their training convergence is slow due to margin ranking loss and random negative sampling. To overcome these challenges, we propose Hyperbolic Graph Collaborative for Heterogeneous Recommendation (HGCH), an enhanced HGCN-based model for collaborative filtering that integrates diverse side information into a heterogeneous collaborative graph and improves training convergence speed. HGCH first preserves the long-tailed nature of the graph by initializing node embeddings with power law prior; then it aggregates neighbors in hyperbolic space using the gyromidpoint method for accurate computation; finally, it fuses multiple embeddings from different hyperbolic spaces by the gate fusion with prior. Moreover, HGCH employs a hyperbolic user-specific negative sampling to speed up convergence. We evaluate HGCH on four real datasets, and the results show that HGCH achieves competitive results and outperforms leading baselines, including HGCNs. Extensive ablation studies further confirm its effectiveness.