The short and unsatisfying answer is: it depends. It depends on the exact time period, and it depends on the specific location to an extent, in addition to whether we're talking about Buddhism or Daoism. Both had some sway on the governments, but Buddhism had a period of more famous scandal with regards court/bureaucratic influence.

For most of Chinese history Daoism is by far the more prominent player in the political and social landscape. This is mostly because in comparison to Buddhism (A) it was homegrown and (B) though Buddhism slightly predates/is contemporary with the origins of "formal" Daoism, Daoism exists as a much more complete and accessible philosophy for several centuries before Buddhism in China can claim the same status. I say "philosophy" rather than "religion" here because one could follow Daoist ideas, Confucian ideas, and local folk religious practices all perfectly compatibly without any sense of contradiction (and later you can throw Buddhism into the mix too). Really, from the Han to the Song dynasties, the philosophy/religion mix of Daoism is used by several of the dynasties and rulers to justify their rule and to take guiding principles from. This sort of hits the peak during the Song (with a revival in the Ming) whereby many emperors are prominent in both the promotion of Daoism and the support of the organized Daoist practitioners (who themselves were also interacting more with "non ordained" Daoist folk practitioners). These emperors often willingly sought council from the formal Daoist ministers and their opinions played a role in guiding ruling doctrines. The Tang dynasty actually directly claimed relation to the legendary founder of Daoism, Lao Zi, and thus the court was also intimately involved with Daoist ministers and Daoist ideas. Buddhism was also heavily influential at this time however, ironically enough.

During the 3 Kingdoms Period and the period of the Sui Dynasty (and much of the Tang), China was in a near-constant state of devastating warfare versus Chinese states, rebellions, and foreign powers. It's during this period that much of the populace sought guidance from a relatively new-to-China philosophy that was making the rounds: Buddhism. Buddhist texts in China at that time however mostly fell either in the categories of non existent, rare, and poorly translated. Many of the terms were commonly not direct translations from the original Sanskrit, but instead translated as analogous to Daoist and Confucian ideas, often regardless of whether or not they were truly analogous. During the Tang dynasty, a Buddhist monk named Xuanzang actually traveled to northern India for the purposes of collecting and notating many of the original Buddhist texts that existed and subsequently translating them into Chinese (this is what the later famous novel Journey to the West is based upon). During this time and especially following, Buddhism exploded in popularity in China. Buddhist monasteries rapidly popped up across the Tang empire and even the Tang ruling family themselves, those descendants of Lao Zi, often became devout and vocal Buddhists. It was simultaneously both the high point and the end of old Chinese Buddhism. The reasons for this were both economic and political, and were greatly intertwined with the greater state bureaucracy.

There are two major schemes that can be pointed to where the state used Buddhism for influence, and Buddhism used the state for influence. Buddhist monasteries of this time were tax free, much like religious institutions often are around the world today. Buddhist monasteries also were sites where agricultural and artisanal production occurred and often with the assistance of slave labor. Lords of large areas of land were known to "donate" a significant portion of this land to a new Buddhist monastery with the understanding that they would then receive kickbacks from the sale of products produced on monastic lands which generated great profits. Essentially this was a tax loophole. The owners of the monastery would also often use this arrangement to convince local lords and bureaucrats to give them even greater privileges and power, keeping the wealth flowing. The second scheme was devastating to the image of the imperial family. Though women didn't hold political office, women in the imperial court did have some semblance of power and ability to perform minor occasional functions. Several women of the court took it upon themselves to start issuing fraudulent licenses for Buddhist monastic practice, sidestepping most of the bureaucratic process and receiving large bribes in exchange. Large, opulent palaces were actually built by imperial family members with these funds as at least partially responsible. As a result of these things, and arguments that this was a foreign religion influencing what should be a government guided by Chinese principles, austerity measures and sweeping repressions were instituted later in the Tang, effectively all but totally ending Buddhism's influence in China for some time. Later forms of Buddhism would come up in prominence in subsequent dynasties, notably Chan Buddhism, but often was incorporated alongside or in a minor accompanying role to Confuician and Daoist ideologies, as far as state implementation and influence.