By taking over city-states in Central Asia, the Han under Emperor Wu were taking control of what transregional route?

He wanted to take control of the Silk Road.

Explanation:

Emperor Wu, or Wuti, or Liu Che, was the sixth ruler of the Han Dynasty. From 133 BCE he launched attacks on the Xiongnu nomadic people, the major security threat from the north for China at the time, and spent many efforts and time to expand the empire in all directions: Southern China and Northern Vietnam were conquered, and Northern and Central Korea were brought again under Chinese control. He also conquered Fergana (in present-day Uzbekista, Central Asia).