eBay UK has recently changed the way verification works there -- you may be running into limitations on the number of times you can go through verification.
Or you may be running into a limit built into the eBay app specifically.
Try signing in using a browser to see if that allows you to reach your account without using the eBay app.
You may want to try to minimize the number of times you need to sign out and back in; if you have more than one eBay account, rather than signing out to switch back and forth, consider using a separate browser for each account so you can stay signed in to each account at the same time.
