Original:  Zheng hoped to restore the Ming dynastic rule on the Chinese mainland , when the Ming remnants' rump state in the southern China was progressively conquered by the Manchu-led Qing dynasty.
Revised:  Zheng hoped to restore the Ming dynasty on the Chinese mainland , when the Ming remnants' rump state in the southern China was progressively conquered by the Manchu-led Qing dynasty.
Edit intent: clarity