Event detection (ED) seeks to recognize event triggers and classify them into the predefined event types. Chinese ED is formulated as a character-level task owing to the uncertain word boundaries. Prior methods try to incorporate word-level information into characters to enhance their semantics. However, they experience two problems. First, they fail to incorporate word-level information into each character the word encompasses, causing the insufficient word-character interaction problem. Second, they struggle to distinguish events of similar types with limited annotated instances, which is called the event confusing problem. This paper proposes a novel model named Label-Aware Heterogeneous Graph Attention Network (L-HGAT) to address these two problems. Specifically, we first build a heterogeneous graph of two node types and three edge types to maximally preserve word-character interactions, and then deploy a heterogeneous graph attention network to enhance the semantic propagation between characters and words. Furthermore, we design a pushing-away game to enlarge the predicting gap between the ground-truth event type and its confusing counterpart for each character. Experimental results show that our L-HGAT model consistently achieves superior performance over prior competitive methods.