edit and read added note with markup tool on pdf file

If you chose File menu > Print… > PDF > Save as PDF, then you have flattened the annotation layer into the main PDF and removed the ability to access any annotation elements.

If you didn't do this, then you should be able to access or move any prior annotation element such as a note, and single-click on it to open and access its text. The one caveat is that if you are using multiple PDF readers and mixing their annotation tools that may lead to some dropped or inaccessible annotations.

On Sonoma 14.5 with Apple's Preview, I have no issues adding, changing, or moving annotations from the markup options — even after the document has autosaved and I have closed the document, quit Preview, and then reopened the PDF document.

Open the Preview Inspector (cmd+i) tool and click on its lock icon to see user permissions on that PDF. Anything that does not appear green such as Adding Annotations or Signatures, or Changing Document Attributes?

Is this PDF local to your hard drive, on a network share, or is it located on a Cloud service?

After entering that Note annotation, entering text, and then clicking outside of it, that should give you what you see in your screen capture. Moving a pointer over that Note annotation should change it to a hand shape to move it, or by clicking with the hand shape on the Note, it should just open.

I doubt I can be of further help as I cannot reproduce the problem here. I am out of ideas as to why this is happening on a local drive and you have not flattened the PDF annotation layer.

Does this problem persist with the same PDF after a reboot?