Edit: I do not believe bullets have an actual effect on the robots, so using EMP bullets should not matter.
Edit 2: Escort drones have been removed, so I would use crawler instead.

Hit PLAY and navigate these menus.

Feel free to skip this part if you are familiar with how the mission works and what to do.

This map details the route you will take. The first circle indicates where you spawn. Subsequent circles indicate key locations.
Click the images to enlarge them.
 
It is important that you kill all sentinels at the start before you begin the EMP. If you do not, you will be locked out of the area (as shown in the screenshot) and further sentinels will NOT spawn if the first few sentinels are still alive (your useless AI teammates will not kill them).
 
This next map details where the sentinels will start spawning. It is recommended you stay around the area that is boxed with an asterisk (*). The first waves of enemies will spawn in this area. After a pause in the sentinel spawning, the area with the text "Rubble" will blow up and a few sentinels will spawn there. Kill those sentinels, and shortly after, the other box above will blow up, spawning more sentinels. When this 2nd explosion occurs, use your escort drone and prepare your killstreaks. Wait a moment for a large batch of sentinels to spawn (this will happen when the EMP is about to go off), and start spamming your killstreaks and EMP launchers on them. This large batch will spawn in the first area that is boxed with an asterisk that I first covered. When using your blade drone, aim downwards so it explodes upon impact, rather than flying up and giving you control of the blade drone wasting your time. Be careful not to kill yourself with your killstreaks! If you are only trying to level up your primary weapon, then use only your primary weapon. Do not bother with any other weapons or killstreaks.
 
Lastly, the sentinels pose little to no threat. Even if you manage to die, you will respawn (but it is obviously better if you do not die). However, you should prioritize this enemy shown below (the Bomber). It will try to blow you up, dealing lots of damage. This is why you have the Blast Protection perk equipped!
On the left, there's a box with a green magazine hovering above. That is an ammo box which sentinels can drop upon death, so do not worry about ammo!

Mission: EMP takes about 3-5 minutes to complete. In addition, the ending of the mission is pretty buggy. Sometimes it gives you the results screen, sometimes it tells you that you have been disconnected from the game. Do not worry! You will still get your XP.

This is how much XP you can gain! The higher your score, the more XP you earn. Repeat the Mission: EMP and you will be leveling up your primary weapon or battlepass in no time.

Thank you for reading my guide! I hope you found it helpful. If you have any questions, you can ask below and I will answer!