If one of your provinces belongs to HRE but your nation is not a part of HRE and you make that province leave the HRE then the following effects would occur
- Relationship with Emperor -50
The province must be a core province to enable the ability to make a province leave the HRE.
Not sure if it stacks.
Having a province within the HRE increases your Imperial Authority. This in turn allows you to pass reforms if you ever become the Emperor of the HRE further down the line.
Any provinces bordering the HRE (including 1 sea province) may ask to join the HRE unless your nation is more powerful than the current Emperor.
If your capital is a member of HRE you become a prince of HRE and therefore can be elected Emperor. This has it's own benefits.
If you're annexing HRE provinces then it is best to core them as soon as possible as the Emperor will soon declare them "unlawful territory" and demand there return when not cored. If you refuse you get -33 relations with Emperor, -10% production and -50% manpower within the province concerned.
This is actually a confusing issue because the DLC expansions wildly change the mechanics for Terra Incognita, and there is very little documentation about this on the wiki.
'Vanilla' EU4 Terra Incognita Discovery Mechanics
In 'vanilla' EU4, you discover new ocean provinces by moving a fleet with an Explorer on to the unknown ocean tiles. You also randomly reveal coastal land tiles when you move into an adjacent ocean province. To normally reveal land tiles, use an army with a Conquistador. An army with an a Conquistador can also directly disembark from a fleet containing transport ships to a coastal province with Terra Incognita.
DLC Expansion EU4 Terra Incognita Discovery Mechanics
If you have certain expansions enabled (El Dorado), the mechanics are almost completely different. Explorers can no longer move into Terra Incognita tiles at all. Instead, any group of 3 or more Light Ships or Transports will have a "Send Explorer" option which shows you a list of possible 'Exploration Missions'.
There are two different types of Exploration Missions. One is "Explore the waters of X", which sends the fleet out to systematically reveal ALL ocean provinces in the 'X' region. The second type of mission is only available after the first Exploration Mission is completed (or partially completed), and it is "Explore the coast of X", which sends the fleet out to systematically reveal ALL coastal provinces in the 'X' region. Which missions are available to you depends on your Colonial Range from one of your provinces to the region in question. You need a certain range to enable the first mission and a bit higher range to enable the second type of mission.
Comparisons and Conclusions
There are pros and cons to each approach. With vanilla mechanics, there is no limit to how far you could reveal Terra Incognita. You could take your very first fleet directly from Japan to Africa and disembark and conquer the Kongo. With the DLC expansion mechanics, this is impossible. However, you suffer no naval attrition damage when on an Exploration Mission. Some missions, like "Explore the waters of Australia" or "Explore the waters of Oceania" would take many, many trips for an Explorer to complete without the DLC expansion mechanics enabled. Additionally, unless you are using the disembark method, it can sometimes be hard to reveal the coastal land province you want using the vanilla mechanics, since the provinces you reveal are completely random (and not remotely comprehensive).
In my opinion, neither version is superior. I play with the DLC expansions enabled because the other mechanics they provide are excellent. However, if you were trying to do some very specific early exploring (for the purposes of pure conquest), I could see how the Exploration Mission mechanics would be limiting, since they are based on the premise that you would want to colonize the regions in question.
A Note About Full World Exploration
Note that there is no way (with either mechanic) to reliably discover uninhabitable land patches of Terra Incognita. You have a chance to discover them over time if you have a province near the area (or an ally does), but it's pure luck. It makes getting the World Exploration achievement annoying when you have 99.9% of the world explored and are only missing a patch of uninhabitable land in Siberia.
Best Answer
It comes from the global pools.
With the exception of the reinforcement bonus who comes from thin air.
(if you had 20 % reinforcement bonus, only 100 unit will be removed from the pool but 120 will be reinforced).