As we see in episode 2: khubilai Khan is faced with the decision wether to kill the chinese emperor or send him to a monastery. Upon receiving the death penalty, the boy runs to Khubilais arms telling him he don't want to die and what folllows is Khubilai suffocating the boy in his arms.
Is this even close to what happened?
No. Though I haven't watched season 2 yet, it doesn't sound anything like what happened in reality. Nothing in season 1 was correct either so that's no surprise.
There were three boy emperors as the Song dynasty fell apart under attack by the Mongols.
I'm not aware of Kublai hugging any Chinese emperors to death..