I think you have a couple of issues here, Bruce:
Issue One: Stacking order. Illustrator wants to select the top-most path. And I'll bet you a nickel that the oval for the eyeglass lens is somehow stacked above the curve for the highlight. When I run across this situation, it's generally because I brought something over the top in the stacking order because I wanted to hide the endpoints of the "highlight" I stacked underneath.
There are two ways to fix this:
a) Select the path, then use the Object>Arrange>Bring Forward or Bring to Front menu command to bring the underlying path higher in the stacking order above whatever's overlapping it. Fix it the way you'd like, then use Object>Arrange>Send Backward to drop it down in the stacking order until the path is again overlapped by the desired Illustrator elements. Easy, but tedious.
b) In later versions of Illustrator, go to the Layers panel and select the sub-layer (element) that contains your intended path, then select the target for that sub-layer/element and adjust it the way you'd like.
Issue Two: Magnification. If you find it hard to select the endpoint you'd like to continue, zoom in on it and it can make it easier to see/select/differentiate between the path/anchor point(s) you want to select and the one you don't. You may lose some context as you can't see
When you account for these two issues and you work your way around them, I'll bet you can lay your pen tool on the anchor handle(s) you want to hit more easily.
And don't worry about it — I won't hit you up for the nickel.