I have played this commander on Brutal and B+ for a long time, earned about 90 ascension levels from them alone and i find the strongest variant to be P0. I am not going to talk about P3 everyone already knows how bad that one is, but i think they all kinda suck.
When playing P1 you trade away from your already cheeks late game to get some “ease of use” on the mag mines that does nothing to compensate for the lack of Horner’s tanky air units. It feels like a trap prestige for people who don’t know how to use the mine arming aggro mechanics to their advantage in the early game. With H&H you can always guarantee early wave clears if you place 2 sets of mines to arm when the wave is announced and then drop a set in front of the active mines right before the wave hits. Inactive mines take aggro, so the inactive front set holds the enemies still while the active mines clear the wave. One place where P1 mines do have an advantage is in base assaults, where you can drop them to clear whatever armies are in the enemy base. This is really the one thing that you can’t do with P0 mines, but they are technically still useful as a distraction tool.
Mines have a huge AOE radius but weak damage (50), which means they can’t be relied on to clear hybrids or T3 units. This is a problem, since your army will mostly consist of Han’s units and you will lose a lot of them to AOE. The enormous cost of Horner’s units will also mean that you are forced into mass reaper to protect your precious galleons and the few ravens you have from enemy air. The reaper death effect is not that good so you don’t benefit much from the improved on-death effects, but it does come in handy for hellions death effect and that is easily the strongest part of this prestige. Unfortunately it doesn’t result in that much more dps compared to P0 because you lose out on the significant others attack speed bonus which can be anywhere between 30-60% (with mastery) on Han’s units depending on what comp you prefer to play. So you get more attack speed (when hellions die), easier to use mines, but your army is made of paper and you have to rely on calldowns to soften even regular attack waves so you don’t lose 50-100 supply worth of minerals every minute. Its no good.
P2 is all about trading away your ok/good mid game for a ok/good late game. This prestige struggles with mineral usage. Because of the slow rate of production, building Han’s units in the early game is not that good, you really need a critical mass to overcome most forces and are generally better off building wraiths in the early game. But wraiths are much more vulnerable on this prestige because you don’t have Han’s army to give them the benefit of significant other, you also don’t have a big ground army to take aggro on enemy ground AA and you also don’t have a swarm of interceptors to soak damage for them. You can usually get by with 2 carrier galleons in the early game although it is notably weaker than the 5 on P0, but the mid game is much more of a struggle. You are liable to lose wraiths on base assaults and Horner’s army cannot take multiple consecutive fights without a break to heal. You can technically go for BC’s instead, they are much tankier but the mediocre firepower doesn’t justify their cost early on. Part of the issue is that even with the vespene discount, Horner’s units are still too expensive and maxing out takes forever on this prestige so the theoretically “better late game” only happens on REALLY long maps, like temple or mist. So you trade away your early/mid game for late game power that rarely comes into play. Its no good.
It really feels every aspect of H&H was designed around the strength of “Significant others”, when you pay for Horner’s units you are paying for a big health bar that gives Han’s units more attack speed, which sucks for P1 because you really need that health bar to tank for your paper army and it sucks for P2 because aircraft are overpriced for the lackluster firepower because they are balanced around the upgrade.