Are there any relationship between China’s Kazakhs and Russia’s Cossacks?

No, Kazakhstan is a country of the Kazakhs. The Kazakhs followed the Uzbeks into Central Asia from the north in the 15th century, and later formed an independent regime in today's Kazakhstan. Later, it was divided into three khanates from east to west: the Great Khanate, the Middle Khanate and the Small Khanate. They had constant relations with the Mongolian Junggar Khanate in the east and the later Qing Dynasty, and the Torgut Mongols and Tsarist Russia in the west. All three khanates were eventually annexed by Tsarist Russia.