Try staying in the system upon completion of the mission for a second AND HAIL STARFLEET. Go to the Dialouge of the specific mission-giver and DIRECTLY ACCEPT CREDIT (and reward) from that character. Then you can leave the system. I don't know if it matters to wait for a BOff to ask you to depart or if you can just go on your own, though.
Also, if you're in a group, make sure you've actually "accepted" the specific portion of the mission you're on.
For example, if you have to beam down to a planet, and a teammate beams down first, make sure you have your science officer or whomever give you the ground objectives, even if you have to click the little "OPS" off to the side if it doesn't pop up automatically.
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For example, if you have to beam down to a planet, and a teammate beams down first, make sure you have your science officer or whomever give you the ground objectives, even if you have to click the little "OPS" off to the side if it doesn't pop up automatically.