In order to understand the POD, you probably need to read those two articles:
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http://www.tofugu.com/2014/09/15/the-island-inbetween-tsushima-and-the-so-family/
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http://www.tofugu.com/2014/09/22/the-in-between-island-tsushima-and-the-so-pt-2/
Reading them tells you that Sō Yoshitoshi, the daimyo of Tsushima, presented the Korean court in April of 1590 with the parting gifts of two peacocks, a spear, a sword, and the first musket to come into Korean possession. The Sō family is highly interesting.
The question is what if Koreans replicated Yoshitoshi's gun and had some of them in reserve before 1592 invasion of Korea?
Nothing significant. Within Korea, domestic production of gunpowder had occurred since 1377, although a variety of weapons had been in use for centuries before then. By the outbreak of the Imjin War in 1592, a large array of cannons and
hwacha were already in widespread use, which would have been more suitable in order to defend fortifications.
In any case, firearms were initially introduced in 1589, not the following year, and were obtained after Joseon envoys visited Tsushima, not the other way around, after which they were stored for further inspection. Analyses of the war by the Joseon court at the time also indicate that Japanese firearms had significant issues with reloading, allowing Korean bowmen to fire more arrows, and although less training was involved with the former (allowing more soldiers to use them), other gunpowder technology utilized by Koreans meant that most Japanese advantages were generally negated after initial advances. Two to three years of analyses and production would also barely have made a difference on the battlefield, as it took the Japanese over half a century to systematically develop firearms through a steady repetition of trial and error. Even though Korea actively attempted to learn how to produce and utilize firearms on a significant level since 1593, due to the assistance of Chinese generals/craftsmen and Japanese defectors, they ran into significant production problems, and it was not until the second invasion in 1597 that the weapons in question began to be used widely by Koreans in battle.
Additionally, the main issue concerning Korea during the war was that fortifications in the southern regions were generally inadequate, while the vast majority of soldiers were stationed in the north. The Imjin War was the first and last (out of around 15 historically) conflict involving an invasion from the south, which explains why the defenders were much more prepared during the second invasion, as they were able to learn from their mistakes. The amount of soldiers available for mobilization had also deteriorated greatly from ~1 million to 100,000-200,000 due to general peacetime conditions for 200 years. The only way this could have been butterflied away is if the Sengoku Period had been extended by 20-40 years, which would have eventually allowed Gwanghaegun to raise a larger well-trained army, specifically in order to confront the Jurchen to the north. However, a delayed Japanese unification might have also made it more likely for the archipelago to conduct cordial trading ties with the peninsula, butterflying a major conflict.
EDIT: One of the first successful uses of arquebuses on the Korean side occurred in early 1593, during the Battle of Haengju, in which Ukita Hideie, the Japanese commander, was severely wounded in the chest by a bullet, in addition to various casualties among other Japanese generals. However, the defenders were very low on ammunition, and were forced to resort to boulders, boiling water, and a variety of other weapons, such as cannons,
hwacha, and bows, indicating that numerous variables were collectively present.
There's no real impetus to. Unless the POD is quite early, the Joseon court isn't going to suddenly change their stance and argue for militarisation.
The defence policy was simple - if put to the extreme, it was like the guns-and-butter balance with no guns. Maximise butter, maximise social happiness.
This too.