In terms of political control and cultural influence on the largest number of people over the longest span of time, I think China has a better claim to be the long-term world superpower than does the West.
Again, sadly at their largest point there was the Holy Roman Empire, which was more clearly a superpower than either the Song or Jin. The Jin (Jurchen) supplanted the Liao Dynasty, which was an Empire formed from one Mongol tribe (not all the Mongols like under Genghis Khan).
One of the key elements of a superpower is its ability to unify and act. The USSR dissolved from superpower status when it couldn't stay unified, Russia remained a powerful country (it still is) however it isn't a superpower any longer.
The Holy Roman Empire was very powerful, potentially a superpower (at least in the beginning when it was definitely more unified than the Song and Jin) but the Mongol Empire dwarfed the HRE drastically.
Put it this way, the Mongol Empire ruled over almost as much territory as the British Empire (I think the difference is ~0.4 million km^2), without their feet ever leaving soil. Considering the British Empire contained Canada, the Thirteen Colonies, Australia, New Zealand (lots of smaller islands too) and the British Antarctic Territories, all of which the Mongols couldn't have got to during their expansion. I think the Mongols clearly won.
Defining China as a superpower is, in my opinion, false due to it never unifying into an Empire. When the Tang Dynasty was in power it had the full potential to be the world superpower, and depending on how you look at the world in the 8th century it was.
This part of history is what opinion depends on. At the time the Roman Empire was collapsing, but still held a huge swathe of the Mediterranean. However, the Caliphate ruled over an even larger part of land on the south of the Mediterranean including the whole of the Middle East, however in this case land doesn't necessarily mean power. Many European powers avoided this area of land because it was deemed worthless, the Mongols avoided this area because they decided it wasn't worth the effort, and their few excursions into this region was based on their practice of killing anyone who double crossed them.
I'd say, the Tang dynasty was quite possibly the superpower at the time. It had the politics, culture, military and most certainly the economic strength. This was probably a product of China being densely populated at the time (relative to Europe and especially the Middle-east and Saharan regions), and I believe the Tangs wide adoption of woodblock printing probably helped put their cultural development well through the roof compared to their European counterparts at the time.
I think I just convinced myself the Tang Dynasty was possibly the superpower at the time, this is possibly helped by the fact the Roman Empire repeatedly tried to ban Chinese silk as it was exporting too much money to China.
Moving towards the end of the Tang, despite their political power crashing their economic and cultural growth were still huge. So during the 9th and early 10th Century they were the only country big enough to be the superpower as the Caliphate had collapsed and the Roman Empire had shrunk and western Europe had the lovely Dark Ages.
So between the collapse of the Roman Empire and the Caliphate (if they count as a superpower, due to segregation and poor land types) and the rise of the Holy Roman Empire, I believe the Tang probably held the superpower position due to it being the first unification of the country.
But the Chinese cultural sphere has long been bigger than the area under the political control of the recognized emperor in a dynastic cycle, in much the same way that Rome's influence extended beyond the broadest boundaries ever ruled by a Roman emperor. The spread of scripts and literacy is instructive: there are more pre-movable-type-printing era writings extant in literary Chinese than in all other world languages combined, by a huge margin. And Japanese and Korean (and formerly Vietnamese) have long used Chinese written characters even after developing indigenous alphabets or syllabaries for what are languages that aren't even cognate with Chinese. Many other cultural influences have spread outward from China to a recognizable cultural area known as east Asia, just as different influences spread from Greece and Rome to an area known as Europe. But for most of human history, east Asia has bested Europe in both population and prosperity.