Thanks for the images and detailed description of this issue. Lets try a different approach as maybe a request is being blocked!
Firstly run test in normal Chrome browser to see what happens with the network requests:
* Open your normal Chrome view Dev Tools -> Network
* Navigate to the page with that Add button (Do not click it yet)
* In Dev Tools -> Network clear the request history
* Now click the application Add button
* Was there a specific request generated for the pop up form being displayed? (keep a note of this)
Now repeat steps with Cypress browser:
* Start running your Cypress test with Dev Tools Dev Tools -> Network open
* Put
cy.pause()
in your code just before your code clicks the Add button
* During this pause manually Dev Tools -> Network clear the request history
* Allow the Cypress test to continue
* Was the same request generated for the pop up form being displayed? Does it have a success status code or error status code?