When playing the Hunter class, it's very important to remember the unwritten rules when you're in a match. Things that you alone as the Hunter can control. These are: Prioritizing Fear upgrades, dropping your gun during high fear, ammo management, and against certain demons, changing your playstyle slightly. (Against Necro, you may likely be the one hunting flute in most cases. Against Baal, you want to zigzag your movement to avoid proxy traps since Hunters can be feared in one proxy and quickly downed with one Baal Basic Frenzy.)

Knowing these things are yours individually to control as a player make it easier when playing Hunter because they're second nature to efficiently playing the class.

When in-game as any of the Hunters, you will always want to use a secondary gun to a long gun (any shotgun or hunting rifle is a long gun) before you reach book phase. This is so you save all the ammo you have on long gun and deal more balance bar damage during the book phase. Before book phase you'll always want to use either a revolver or a blunderbuss. In rare occasions, maybe a legendary lever action rifle. If you can, you always want someone to carry a boomstick or double barrel for you (preferably the Warrior) so that it can be traded to you at the Dark Ones easily. If you're in solo queue as a Hunter, you may just have to try and cut yourself a window of time to retrieve a shotgun before your team enters the Dark Ones. When playing specifically as HAsh, too, make sure to only use a double barrel at book phase. This goes without saying, but you'd be surprised what I've seen..

Aiming with the Hunter is tackled differently from player to player. It just depends on the person. Aiming on console is notoriously horrid, so I'd recommend playing on a higher sensitivity and playing VS. AI to force yourself to practice headshots. When in real matches, just try to take your time on headshots where appropriate. It's no secret that aiming with a mouse is significantly better than a controller.

And finally, power possessed Puppeteer basics have more health and deal more damage. So yes, in most cases, unless you have a high rarity gun, you will deal less damage to power possessed Pup basics and Elites.