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The part that moves the cylinder is called a "hand" or a "pawl", depending on brand of gun. S&W calls it a "hand", Ruger calls it a "pawl". Unsure what Hertitage calls that part, but probably "hand".
Unless the gun is just totally worn out, then the hand/pawl spring is probably just broken or over compressed to where it lost it's spring. The spring pushes the hand/pawl forward to engage the cylinder's ratchet, and the cocking of the hammer pushes the hand/pawl up to turn the cylinder. The issue could be caused by something else being broken, but the hand/pawl spring is the weakest link and most likely to fail.