How did you implement the shift so it added the link to the general blog page?
When I click the ‘read more’ link, it brings me to a different page (with more blog posts) – Is that not the desired result?
Depending on your access to files and your experience with code, I would recommend looking over: https://codex.wordpress.org/Customizing_the_Read_More
If you go to Setting → Permalinks in your WordPress Admin, do you see blogging entered in the Custom Structure field (after your site’s URL)? If so, remove it, and click “Save Changes”.
Then visit your site again, and see if the “Read More” links are now working correctly.
Edited to add: after removing the blogging bit from that field, you’ll also want to either enter something like %postname% in its place, OR choose a permalink option other than “Custom Structure”. (Otherwise, WordPress will just put you back to using the Plain permalink settings option when you click the save button.)
@allisonplus – thank you for the information.
@girlieworks – Thank you very much for the detailed instructions. Your suggestion worked 100%. The reason I had the blogging page created was so even if they click the read more link, it would take anyone to an over all page that houses all of the posts I create. Now I have to figure out how to make this happen, but thank you both for your help.