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In Europe, armies were often of generally the same size and makeup (at least in the instances you mention) and tactics codified, so in open engagements equipment and (that being equal) minor differences in proficiency could well mean the difference between winning and losing a battle.
In the Chinese example you mention, sheer force of numbers caused Qing to win the day, that and the slow reload time of the opposing cannon as compared to Qing archers. Simply put, despite the cannon killing more Qing troops per shot than were killed per Qing trooper, they were still wiped out because they couldn't kill quickly enough to avoid being overrun.
The same happened during the Korean war, where DPRK and Red Chinese human wave attacks overcame American and South Korean forces despite the latter having superior equipment.
And paradoxically to your question it also played out in Europe during WW2 where after D-day Sherman tanks were engaging German Tiger tanks. While the Tigers had far better guns and armour, and could kill Shermans before the American tanks were even in range, they were usually fighting alone or in such small groups that the massed numbers of Shermans they were facing could close to within range for their underpowered guns to score a kill before they were all destroyed. The same happened on the eastern front as well between T-34s and Tigers.
Another example is the destruction of 3 Roman legions in Germania by Germanic tribes. The Romans had superior equipment and training both, yet were wiped out to the last man because they could not bring that equipment and training to bear effectively against the assault they were facing (they simply weren't trained for what was effectively guerrilla warfare when the Germanic warriors came at them while they were on the march through the dense forest).
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The first cannon the Chinese had, according to the Portuguese were three (3) cannons, presumably ship cannon, manned by Portuguese crews which the emperor specifically requested of them in 1621. Three cannons are not enough to affect a war in which there are many thousands fighting over vast distances. Also, ship cannons use carriages which are not effective in field warfare, which requires much more complicated and advanced carriages, not to mention a skill level way beyond that of a typical ship gunner.
About 50 years later Ferdinand Verbiest, a Portuguese scientist, constructed about 100 cannon for the Chinese emperor, however, by this time the Manchu had completely won and even then it is likely these cannons would have changed the tide of battle for logistical reasons and the problem of providing and moving around Portuguese gunners. Moving cannon around in those was a non-trivial problem because you have to transport the powder too, and keep it dry. Cannon was most effective in siege situations at that time, which was not the problem facing the Ming.
It was not until over a 100 years later that the Chinese acquired "firearms", once again from the Portuguese, and only in very small quantities. For example, the emperor's guard in 1792 was equipped only with halberds. Even in the Opium Wars (1840) the Chinese carrying muskets were so small in number and ineffective in their operations that they were more or less inconsequential. So, your idea that the Chinese were using "firearms" against the Manchu is not correct.