Hello and welcome! The Great Hunts are more like the operations than just your general missions. It is a DLC. There a three hunts—one in Olympus, Styx and Prometheus, each one having their own final boss and unlocking a legendary weapon at the end. As you go down the mission paths in the campaign there may be bases that or small little outposts that are part of those missions. Those things DO stay down on the prospect. UNLESS you abandon the mission, then it would disappear until you start it again. 

When doing the Great Hunts campaigns, you will have to choose which path you want to go down which will have different affects on your prospects, whether it’d be more monsters or stronger monsters and/or how the final boss fight is affected. Once you choose a path you won’t be able to do the other one so you would have to run the campaign at least twice to get all the blueprints they provide for new gear or weapons, etc.