iCloud Mail connects with your iCloud email address and an app-specific password, not your regular Apple Account password. You can add it in the desktop and mobile apps, but not on the web.
Click your avatar in the top-right corner and choose Add Account.
Choose iCloud.
Enter your iCloud address under Email address.
Paste the app-specific password into Password. Display name (optional) is the name Newton shows for this account.
Click Test connection. When it says Connected, click Connect.
iPhone, iPad & Android
Open the menu and tap the Settings gear at the bottom.
Tap Accounts › Add account › IMAP (iCloud, Yahoo…).
Tap iCloud Mail.
Enter your iCloud address under Email and the app-specific password under Password.
Tap Add account.
Setting up Newton for the first time? Tap the iCloud tile, enter your email address and app password, and tap Add Account. The Open iCloud Mail account settings link on that screen opens your Apple Account page, where you create the password.
Good to know
Apple only lets you create app-specific passwords when two-factor authentication is on for your Apple Account.
Newton fills in iCloud's server settings for you: imap.mail.me.com (port 993, SSL/TLS) for incoming mail and smtp.mail.me.com (port 587, STARTTLS) for outgoing mail.
Newton stores the password encrypted on its servers, so your other devices can connect without asking for it again.
In the mobile app, the Free plan includes two accounts. Adding a third there needs Newton Pro.