Deep clustering with attribute-missing graphs, where only a subset of nodes possesses complete attributes while those of others are missing, is an important yet challenging topic in various practical applications. It has become a prevalent learning paradigm in existing studies to perform data imputation first and subsequently conduct clustering using the imputed information. However, these ``two-stage" methods disconnect the clustering and imputation processes, preventing the model from effectively learning clustering-friendly graph embedding. Furthermore, they are not tailored for clustering tasks, leading to inferior clustering results. To solve these issues, we propose a novel Attribute-Missing Graph Clustering (AMGC) method to alternately promote clustering and imputation in a unified framework, where we iteratively produce the clustering-enhanced nearest neighbor information to conduct the data imputation process and utilize the imputed information to implicitly refine the clustering distribution through model optimization. Specifically, in the imputation step, we take the learned clustering information as imputation prompts to help each attribute-missing sample gather highly correlated features within its clusters for data completion, such that the intra-class compactness can be improved. Moreover, to support reliable clustering, we maximize inter-class separability by conducting cost-efficient dual non-contrastive learning over the imputed latent features, which in turn promotes greater graph encoding capability for clustering sub-network. Extensive experiments on five datasets have verified the superiority of AMGC against competitors.