Bungie released Harbinger, the activity that allows you to farm Hawkmoon with random rolls and get the weapon's catalyst but if you are having trouble finding it, we might be able to help you.
Harbinger is completely focused on Hawkmoon so it's pretty logical that you need to have obtained the weapon in order to kick the mission off. If you haven't, and are unable to start it, you can check out our guide on the prerequisites for obtaining the Exotic Hand Cannon .
After you have the weapon, you can bring it to the mission but it's not a must, except for the associated triumph. Before you can start the mission, however, you need to go to Trostland, EDZ, and go past the church where Devrim is located, into a ruined building that has a fireplace in it. There are many buildings around so here is an image showing which one you're looking for exactly.
Here is a view of the entry point while standing in the church.
When you get to the fireplace, take your Hawkmoon out and shoot at it to break the wall. There is also a possibility that someone else already broke it. Enter through the hole and you will immediately be next to the node that starts Harbinger. Activate it and proceed through the tunnels ahead of you.
When you navigate far enough through the tunnels, you will reach an area with brick buildings. Look for a hole in the right wall and go through it, as pictured below.
Proceed through the hole and you will eventually reach the outside. Stick to the ledges to your left and a Taken Hobgoblin will eventually start shooting at you. Kill it and jump to the ledge where it was standing.
Once you're there, take a look at the dam and try to spot the tree branch from the image below. Jump to it and then jump all the way to the dam's closest ledge.
The ledge can be seen exactly where I'm aiming.
Once you land, go past the first three drainage pipes and turn left. Start climbing those ledges, with the ultimate goal being beneath the scaffolding to the right.
When you get through that door, navigating the dam is pretty straightforward but prepare to be instantly killed, repeatedly. Taken Vandals will do this as well as Savathun's Emissaries, of which you need to kill three. It appears that modifiers for the mission will change each week but when Match Game is active, either bring a combination of three Light elements via Special and Heavy weapons on top of the spec you choose.
Alternatively, you can use weapons that can shift affinity, such as Borealis or Hard Light, in order to break the elemental shields. Arbalest is a great choice as well since it breaks all shields effectively.
Upon defeating the Emissaries, the quest will lead you to a room and a fight that resembles the Lake of Shadows boss fight but you will be facing several champions here. The first week had Unstoppable ones. You will see which champions you'll face when launching the mission to bring gear accordingly.
If you manage to not rage quit the encounter, follow the quest marker to proceed. Eventually, you will repeat the jumping puzzle that you did when chasing Hawkmoon but this time it has a few champions thrown in. You will probably have to cheese the boss fight in the end by picking high ground and thus exploit the inept AI but when you finally down the bullet sponge, you will be awarded two randomly rolled Hawkmoons and the catalyst. Congrats!