To prevent the pen tool (p) from changing its function when interacting with other anchor points or paths, open the general preferences dialogue (illustrator > preferences > general) and check the disable auto add/delete option.
View Add Anchor Point Illustrator PNG. Then, you can simply press the delete key to remove the strays. Even though you can use a hidden tool to delete and add anchor points, illustrator automatically does this as a if you prefer to use the tools dedicated to adding and deleting anchor points, choose edit→ preferences→ general (windows) or illustrator→ preferences→.
How to round an anchor point / line in Adobe Illustrator ... from i.ytimg.com
I think this will work, but if.when your using the convert anchor point tool to create a new handle on a point, as you drag the guide line is visible, but you also still see the old. However, you should not add more points unnecessarily as it makes the path illustrator provides you features to smooth the appearance of paths as well as simplify paths by removing excess anchor points. Ideally, your design should use as few anchor points as possible, so that it's easier to edit on the fly.
Illustrator will scan the entire project for stray anchor points, and select them all.
Adding anchor points can give you more control over a path, or it can extend an open path. However, you should not add more points unnecessarily as it makes the path illustrator provides you features to smooth the appearance of paths as well as simplify paths by removing excess anchor points. I also explain why you would need to add anchor points (as well as how to do it). Switch to the anchor point tool, find the anchor point, put the sharp end of the v on it, hold down the mouse button, and drag.
