My iPhone (App Store app) is not recognizing my hotmail account password and therefore won't let me install an app.

The Hotmail address used as the Apple ID is a Microsoft account. If the password for that Microsoft account has stopped working, the account password must be recovered and reset successfully before it will work anywhere (including the App Store on iPhone).

Follow these steps to fix the password issue on the Microsoft side first:

1. Go to the Microsoft account recovery page: Recover your account.
2. Enter the Hotmail email address used as the Apple ID.
3. Follow the prompts to receive and enter a security code, then set a new password.
4. Ensure the correct domain and spelling are used (for example, hotmail.com vs hotmail.co.uk, and correct spelling of the name part of the email). If the email is not recognized, try any other alias or phone number on the account.

If sign-in still fails and the message "We couldn't find an account with that username" appears, contact Microsoft support from the same page using the Contact support option and explain that the username has stopped working.

Once the Microsoft account password is confirmed working (for example, by signing in at account.microsoft.com in a browser), update that same password wherever the Hotmail address is used, including on the iPhone if prompted.

If the account is configured to use passwordless sign-in with Microsoft Authenticator, it may be necessary to temporarily turn passwordless off and set a password again so that password-based sign-in works where required. This is managed from the Microsoft account security page after signing in.

References:

• My username and password have stopped working
• How can I recover my password for Microsoft accounts, including Outlook.com, Hotmail, Live, MSN, Onenote, etc.?
• Updating your email account password.
• Download Microsoft Authenticator
• Why can't just use my password anymore to log into my email. been going on for a while.... - Microsoft Q&A