Cannot connect to AppStore

I have not seen a definitive solution for your problem. So, I suggest you use my search link in trying to solve it: authentication verification failed apple. You will see a number of links that are, or may be, related to your specific problem. Several links related to my search claim fixes.

In this regard, see See Get a verification code and sign in with two-factor authentication https://support.apple.com/en-us/HT204974  Below is an excerpt, but read the whole document. See the bold italics below.

"Whenever you sign in with your Apple ID on a new device or browser, you'll confirm your identity with your password plus a six-digit verification code. There are a few ways you can get a verification code. You can use the code displayed on your trusted device, get a text or phone call, or generate a code from your trusted device [see the additional information in the article]. 

A trusted device is an iPhone, iPad, or iPod Touch with iOS 9 and later or a Mac with OS X El Capitan and later that you've already signed into using two-factor authentication."

If you want to contact Apple Support to deal with your problem, see:

Get a call from Apple support to help solve your problem using this link https://getsupport.apple.com/?caller=kbase&PGF=PGF63005&category_id=SC0245&symptom_id=23362 Be patient. Sometime this site is not fully functional.

Apple ID Support https://support.apple.com/apple-id

Apple Support Twitter https://twitter.com/AppleSupport  There may be a wait of an hour or so.

Apple ID account security for country-specific and region-specific help https://support.apple.com/en-us/HT204169

Get Support: We’re here to help https://getsupport.apple.com/. Go through the options to find the best one.

See if anything here helps.

App Store, iTunes Store, or other Apple services - If you can't connect

Have you tried another Internet source (cellular instead of Wi-Fi for example)?