(Having got) what they wanted,they won't come back.

这个句子可以理解为独立主格句型.首先把它还原成两个分句,
They have got what they wanted,they won't come back.这个句子是错的,因为没有连词,所以考虑独立主格,因为they 和 have got主谓关系,所以用-ing.-ing表示主动,-ed表示被动.
祝学业进步!