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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