My horse doesn't neck rein so as we approach the barrel he works best when I tip his head in lean in bump hard with the outside leg and push my heel into his inside the pull his head around but then again that's just me
I also use inside and outside leg. (easier to describe than right and left. Inside is in the direction you are turning) I cue for a turn with my outside leg and my inside leg I use as a pole for him to balance and turn around. It doesnt allow him to drop his shoulder or into the turn. It also keeps him rounded. I rein, either direct or neck, lightly and as necessary to keep him forward and flexing.
Of course... I was a barrel racer... that's just the way I was trained and do do it...
Its kinda hard to figure out what you have to do unless you have the money for a trainer where I didn't when I first started in the rodeos but I figured it out I just spur both sdies with equal pressure and then I lift the inside rein up towards my shoulder instead of pulling back cause that can do one of two things 1 make your horse slow down and 2 make him drop a shoulder into the barrel.