So it could either be the head that moves not enough or the eyes that move too much.
First let’s make sure the head moves enough. We’ll look at a few vertices and confirm that they’re weighted 100% to the head bone. For this, get into edit mode (or weight paint mode), select any one vertex on the head, and look into sidebar ->vertex weights. If those show a weight of 1 on the head bone, the head is fine and it’s the eyes that are problematic. If instead they show a total weight that’s split between the head bone and some other bone(s), you found the issue.
The eyes then : what could cause them to move further than the head ? could be that they’re parented/constrained on top of being weighted to the eye bones, could be a modifier… please check all of that.