First, I must apologise in advance for any poor formatting. I am currently without access to my laptop so this will be the first answer I have had to write on mobile.
With that out of the way, you’re not wrong to draw a link between New World silver and the Chinese economy. The former did have a significant effect on the latter. But before we continue, I must make two declarations: firstly, I will, as usual, focus on the Qing Dynasty during the 19th century as my reading has predominantly been in this period. Secondly, my familiarity with Chinese economic history extends largely as it pertains to political affairs, especially the Opium Wars. As such my answer is built primarily around one contemporary work, Man-Houng Lin’s China Upside Down: Currency, Society and Ideologies, 1808-1856 (2006), although as this remains a standard work in the economic history of the Late Qing this should be representative of the general consensus (notwithstanding more recent studies in Qing numismatics which have added an extra layer to the silver-copper coin problem.)
Lin argues that we should see 1775 or thereabouts as the point at which China effectively became near-totally reliant upon silver from Latin America. Provincial records show that Guangxi and Yunnan, the two provinces with significant silver mining operations, ceased to be even self-sufficient, let alone able to supply other provinces, around this time; and Japanese silver exports had sharply declined since the founding of the Tokugawa Shogunat, with barely $140k shipped from Japan between 1709 and 1761 compared to $68 million reaching the Philippines from Latin America between 1721 and 1740. Even accounting for the fact that not all silver going to the Philippines then went on to China, even by the mid-1700s it is clear that the New World would become China’s main source of foreign silver. Indeed, those provinces that did rely on foreign silver even switched to using Spanish (and later Mexican) dollars in place of local sycee ingots due to their stability and perceived legitimacy. Let us not, however, suggest that there was a state of dependence just yet. By the beginning of the 19th century it was st ill only the maritime trading provinces of Guangdong and Fujian that made significant use of Spanish dollars over ingots.
The important thing to note is that China used bronze or copper coinage alongside silver ingots, and whilst this was a system that worked well in fair weather the problem was that copper minting was not regulated like for silver. As such, copper coins might be minted at varying rates regardless of whether there was a dearth or excess of silver.
Thus, if the introduction of New World silver had an inflationary effect then this would have been reflected by an increase in the relative value of copper. This, as it turns out, didn’t really happen. The exchange rate of copper to silver fluctuated between around 850 and 1000 copper cash per silver tael for most of the 18h century, in fact hitting a peak in 1794 of 1150 (suggesting either an inflation of copper or deflation of silver) before returning to around 950 by 1805. It seems, then, that the Chinese system was operating pretty normally.
Until the Napoleonic Wars, that is.
Political turmoil in Spain slowed the production and movement of silver from Mexico, and with Mexico declaring independence in the 1820s silver output slumped further. By 1816 the exchange rate reached 1,177 cash to the tael. This became 1,271 by 1826, 1,487 by 1836, and reached 2,208 by 1846, hitting a peak of 2,355 in 1849 before finally declining again over the next two decades. The Mexican independence wars are estimated to have caused an over 50% slump in global precious metals production compared to 18th century figures, and the near-halving of relative copper value in China almost mirrors that exactly. So in a sense New World silver did cause inflation in China, but by its absence rather than its presence, and only to bronze coins as a result of deflating silver.
Not OP but I have a follow up question. How much silver extracted or money from silver extracted went to the Qing empire?