Cannot connect to iCloud on my iPod touch - Apple ID is not accepted

This is an old iPod touch. Model PC540LL/A. The "About" screen shows version 6.1.6

My kid uses it to listen to music and podcasts. It used to work fine but recently it has been asking for the Apple ID to be able to connect to the iCloud services. However, the system does not accept the Apple ID. Web searches indicate that I may have to append a verification code at the end of the password, but I have no way of obtaining such code.