I Can't Sign Into My iPod Touch 4th Gen.

On a newer iOS device, go to Settings => your Apple ID => Security => Get Verification Code

Then enter the code after your password (no spaces) on your 4th Gen

Hope this helps

Re: "I Can't Sign Into My iPod Touch 4th Gen : It seems I need to use 2 factor authentication.

I am not getting any verification codes to my iPhone, so how can I sign in?"

Check you do not have "Do Not Disturb" turned on, and that Notifications are allowed on iPhone.

You may want to: Check your Apple Account device list to find where you’re signed in - Apple Support

and

See how to: Get a verification code and sign in with two-factor authentication - Apple Support

Eg: says:

"Use the code displayed on your trusted device

When you sign in, the verification code is displayed automatically on your trusted devices.

1. Sign in to your Apple Account on a new device or browser.
2. Look for a sign-in notification on any of your trusted devices.
3. Choose Allow to receive your verification code.
4. Enter the verification code on your other device to sign in."

________

If needed: You might want to look through the iPod's User Guide for tips:

iPod touch (4th generation) Manuals and Downloads - Apple Support