To archive a mission, all you have to do is go to your 'missions' tab and select the mission you want to archive. Click 'mission options' and select 'Return Budget left'. You will see a green notification in the top right of the screen telling you your budget has been returned. Click the 'missions options' button one more time and select 'Archive'. You will then get another green notification telling you the mission has bee archived.

See how here:

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