Do you own the season 12 season pass? The Hawkmoon mission requires ownership of the season 12 pass. You can check if you own it by pulling up the Director and inspecting the new Timeline icon on the main map. There should be four small nodes under Beyond Light representing seasons 12 through 15. If you hover over the first node it will show Season of the Hunt and if you own it it will state Onwed at the bottom of the info card.
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I followed the instruction you had posted and I don’t have the season 12 pass and when I do click on “Learn More” it prompts me to buy the Deluxe Edition for $30. Wasn’t sure if there was a way around this or not without having to pay for the Deluxe Edition
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Bungie has only ever sold the season pass for the current season. There is no way to purchase individual season passes for past seasons. The only workaround is by purchasing or upgrading to the Beyond Light Digital Deluxe Edition which comes with season passes for 12 through 15. That's why the link takes you to the Deluxe Edition.