Shoulder influence is not talked about very often but really should be.
In the most basic way the ability to influence the shoulder is required to be sure you can ride your horse on purpose from point A to point B. And maybe it would be easier to think of shoulder influence as front end influence.
When riding the horse it should be evenly bent from the tail to the ears. When you are riding that can be really hard to judge. The most common mistake is not having influence on the horse's neck or shoulder area. The horse either over bends its neck to the inside or counter bends.
If the horse overbends to the inside, it will feel like its shoulder is stuck on the wall and you will have a harder time starting a circle. On the otherhand if the horse is carrying his neck to the outside the rider will talk about the horse falling in on their inside leg.
I have actually stoped asking riders to bend the horse's neck. I have actually started asking them to flex the horses neck (about 1") and then if they need to turn the horse's front end to move both reins with equal pressure to the direction they would like to move the horse. This really helps if the rider then also pushes the horse with their legs. Really the reins act as a frame in and the legs push the horse into the frame the reins have established.
If a rider overbends the horse to the inside or counterbends the horse to the outside the shoulder is free to do whatever it wants and more often than not - the shoulder blocks the goal of the rider.
The hint is ride the whole front end of the horse for better shoulder influence!
Enjoy your horse every single ride!