Do you mean the reputation projects for the summer event? If you go to the rep screen, you should see a little "cancel" button on the project, where you'd normally find the button that says "collect reward." This is something the devs added recently so you can cancel projects you don't want to finish, or selected by mistake. Keep in mind, though, that you'll lose everything you've put into it so far--you don't get a refund.
If you're talking about a PvE mission, you can go to the "In progress" tab under the mission screen and select the mission you want to cancel. On the right you'll see a series of buttons--Make Primary, Hail and Drop, I believe. If you click on Drop, you'll cancel the mission.
I looked and I do not see any button saying cancel all it shows is progress the quests I have are the risa one were you have to get pearls and the crystal enty were you need to get shards
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If you're talking about a PvE mission, you can go to the "In progress" tab under the mission screen and select the mission you want to cancel. On the right you'll see a series of buttons--Make Primary, Hail and Drop, I believe. If you click on Drop, you'll cancel the mission.