Took me a few tries, but looks like fewer than most of the people here (it helps that the last few Privateer missions were obnoxious enough that I pre-looked it up when I had the name when I selected it). What were they thinking, giving me suicidal allies for the Wolf one??!?!).
Firstly, I want to throttle them for wording it this way. This was my last Privateer mission, EVERY OTHER TIME they referred to this as "weak points", and now suddenly it's "firing on their powder stores"? As far as I recall, the "weak point" thing was never explained as such.
The first "fleet" is no problem, of course; 2 shots each from the pivot gun (left trigger instead of right. I'm on Xbox 360, if the button assignment is the same on the PlayStation, L2 instead of R2), or try to line up at least 2 for Grape Shot. I find the one on the left is a loner, so I sail between him and the rest, getting him early with the Pivot, then Grape Shot and Pivot for the rest. On to the real problem.
I believe this would be one Man O' War and two Frigates, mostly going by the nomenclature used in Black Flag. At first, they all looked the same to me, but I realized the Frigates seem weirdly short when I see them from the side.
I start by firing Chain Shot at the MOW. I find this is the only weapon I have to fire ahead of them; if I have them perfectly in my sights every chain misses. This sometimes takes me 2 shots. When I'm successful, the game tells me how to change to Round Shot (i.e., the default / first weapon. Automatically starts as Up on my directional pad. Between defaults and my settings, I ended up with Chain to the right, Flame down, and Grape to the left. You can change this in the Weapon Select screen, Right Bumper, or R1). Then I "run off" and focus on the Frigates. Most of the time, this took the fight well away from where the MOW was sitting, so I could freely ignore him.
I find that they can each take ONE barrage of Flame Shot, so I did that. After that, I switched to Round Shot and make sure it at least half misses (you can tell by the white glow on the water when you hold the trigger). After several half-hits, I would get that blessed weak spot.
The time that worked just now, I lucked out, and the second Frigate rammed me at an angle (like in a car chase movie where parallel cars bump each other). I'm not sure if it made a difference, but I dropped to Half Sail at the last second before impact. Then I had the ship in my face with a giant weak spot (and my guys missed, the aiming reticle moving off at the last second! I turned away to not bump him again and got him after the reload).
Finally, to finish off the Man O' War, I sailed up to him and "parked" perpendicular to him (i.e., if he could move, I'd be in the perfect spot for him to T-bone me at a 90-degree angle, though I'm not sure if I was in front of or behind him. He might have forward-facing weapons, so that can make a difference). In that position, he couldn't shoot me. I angled things so that again most of my Round Shot would miss, and several shots later I had my weak spot, then my win!