If you hold at least 20 shards, you will earn extra rewards when you defeat Oryx in the Taken King version of King's Fall.
― In-game description
A Moldering Shard is a material usable in the raid King's Fall. If a player possesses 20 or more Moldering Shards when Oryx is defeated, 20 of them will be consumed and an additional piece of loot will be dropped from the loot chest.
Moldering Shards are obtained randomly by opening chests and killing bosses in King's Fall. They can also be purchased from Eris Morn for 2,500 Glimmer and 15 Hadium Flakes each at Crota's Bane rank 5.
Moldering Shards are neither acquired nor consumed when playing King's Fall on the Heroic difficulty setting.[1]