Could the Song have taken northern China back from the Jin?

Aug 2016
I believe there is a famous Chinese general who fought very well against the Jim but was ultimately executed due to affairs in the Song court. I have also heard that the Song sometimes tried to appease the Jin.

What if the Song were more dedicated to defeating the Jin? Could they have taken back Northern China before the Mongols invaded the Jin?
May 2009
Imagine what a nasty suprise it wouldve been if the Sung finally defeated the Jin and got the north back only to run head-first into Genghis Khan and a unified Mongolia. That wouldve been a very short victory celebration.