Way to unlock the mission: you get the opportunity to start this side-mission right after you leave the police precinct in Burnley and you get to complete during the Escape the GCPD mission.
The beginning of the mission
Right after you return to Burnley (the precinct's rooftop) Barbara Gordon should contact Batman, and ask him to stop the Penguin. There are six stolen crates of weapons that you need to destroy. You can reach them in any order. It is a good idea to use the side-missions menu to determine the next destination, because that is where the locations of all the stolen crates are (the above screenshot).
Destroy the first arms cache
The first cache is in the Eastern part of the Burnley district, at the back of one of the communications towers connected with Enigma. Attack the corrupt policemen from above. Then, continue with more standard modes of fight.
After you have secured the area, select the disruptor from the inventory, target the crate shown in the screenshot and hold down the appropriate trigger/key to compete the process. Listen to Barbara's short comment and locate the next crate.
Destroy the second arms cache
The second arms cache is in the Northern part of the Diamond District. Just like in the above case, it is worthwhile to attack the Penguin's goons from above and deal with one of them at the very beginning. After you start the fights proper, watch out for the enemy with the shield, stun him with the cape and stomp on him. After you have secured the area, select the disruptor from the inventory, target the crate in the truck and use the gadget.
Destroy the third arms cache
The third arms cache is in the Northern part of the Coventry district. Just like before, attack the Penguins goons from above. Right after the main fight starts, watch out for the enemies with shields and the ones with guns. Target them in the first place and, whenever necessary, use the gadgets to your advantage. After you have secured the area, select the disruptor from the inventory, target the crate and use the gadget.
Destroy the fourth arms cache
The fourth arms cache is in the northern part of the Sheldon Park district. You need to reach a small balcony where there are three enemies equipped with guns. You necessarily need to use here the smoke pellet or the concussion detonator to make the enemies more vulnerable to Batman's attacks. After that, try to deal with them before they regain their strengths and resume firing at your character. After you have secured the balcony, select the disruptor from the inventory, target the crate and use the gadget.
Destroy the fifth arms cache
The fifth arms cache is onboard The Final Offer - the Penguin's ship, located in the North-Eastern corner of the map. Since you have already secured that area while completing one of the initial main missions, you should meet here no new enemies.
You may find it a bit problematic to find the arms cache. You need to stop in the spot shown in the screenshot and target the container hanging above the ship. Use the claw to open the container's door and use the disruptor afterwards, to be able to complete this part of the mission.
Destroy the sixth arms cache
The last, sixth, arms cache is in the North-Western parts of The Bowery.
The important piece of information is that you need to make it to the lowest level and jump over the metal fence. Then, reach the tunnel located next to the subway entrance (screenshot 1), where you will meet quite a few enemies. Start the fights, during which you need to watch out for the enemies with knives and not let yourself be cornered. Also, do not allow the armored enemies to pick up the knives, because then they will get really dangerous. After you finish the fights, locate the weakened fragment of the wall shown in screenshot 2 and spread the explosive gel over it. You can now use the disruptor on the last arms cache and listen to the conversation with Barbara.
[New Character Trophy: Penguin Gang Member]
[New Concept Art: Final Offer - Boiler Deck]