这句话前半句有没有错?Hearing her mother had come,she was happy.

前半句无错.
1.hearing her mother had come是现在分词短语做句子的状语.
因为句子主语she和动词hear之间为主动关系,所以用其现在分词形式hearing引导该状语.
2.hearing后面是一个省略了引导词that的宾语从句：(that) her mother had come
从句主语为her mother,谓语为had come
该从句使用过去完成时,是因为come该动作在谓语动作hear和was happy之前已经发生,即“过去的过去”.
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有错误
仍然是多个谓语动词，
这不是主从复合句，因为没有连词，hearing用了非谓语，但是had come仍然要处理成非谓语

had come处理成：hearing her mother's coming,she was happy