I deleted my previous expedition save file. Now when I try to launch an expedition it just says expedition completed and I can't create a new playthrough. Is there a patch I need to download?

Expeditions saves are independent of each other so deleting a save of a previous expedition should not affect the current expedition. I am not sure if you mean that you deleted the save for a previous expedition or one that you just recently started for expedition 7.

When you launch the game and are at the first window the build number is listed in the very low right corner. The build number for 3.90 is 87850.

If you have the right build number then first try a Verify/Repair by selecting the button to the right of Play, then Manage Installation then Verify/Repair. This may take a few minutes and screen will look as if it is updating the game.

If you have a number lower than 87850 then No Man's Sky is not updated and you might want to submit a support request to GOG. They may suggest uninstalling Galaxy only and then to reinstall Galaxy and scan for games.

There has been a bug in some previous expeditions with prematurely converting an expedition to a normal save and it is possible that you are experiencing that type of bug.

Good Luck.