The info I will be giving comes from a course I took on Mongol History taught by Professor Scott Levi at OSU. The most comprehensive book we had to read for the class was "The Mongols and Global History" by Morris Rossabi: https://www.amazon.com/Mongols-Global-History-Norton-Documents/dp/0393927113

In the terms of your question, there was eventually a Mongol capital largely founded by Genghis Khan's successor Ögedei Khan called Karakoram. The Mongols, however, were nomadic peoples that often had little to do with the cities of their sedentary neighbors like China. The Mongols' mobility through their cavalry became one of their greatest strengths during their conquests of Asia. The Mongols utilized large grass/steppeland of which they could graze large amounts of horses and cattle on. Part of why the Mongol invasions were so damaging in places such as in Khorasan was because they attempted to convert the farmland into grazing-land (This portion comes from a class on Early Islamic Society taught by Professor Jane Hathaway).

Certain portions of the Mongol empire, as it decentralized, did become sedentary and have interest in city building such as Kublai Khan's Yuan China which established Beijing as its capital. So it is a false generalization to suggest that they did not build cities, but their often nomadic society changes their relationship to cities as compared to comparable sedentary empires.