Number count for icons gone

Which email account are you referring to is not clear from your post. Google, iCloud, AOL, Exchange, Outlook, Yahoo or any Other

For email, this is called Push / Fetch Settings. You may know that there are two settings (Fetch and Push Mails). What have you set? Watch the GIF below

Google, AOL, Yahoo - Only the Fetch mail Option is available

iCloud, Exchange, Outlook - Both Fetch and Push Mail option is available

1. Push Mail: Whenever possible the email client will push emails to all your devices.
2. Fetch Mail: You need to set How you want it to be fetched, Automatically, Manually, Hourly, Every 30 mins, every 15 mins
   1. Automatically: will fetch when the iPhone is powered and on WiFi
   2. Manually: with fetch when you open the email, open Inbox, and swipe down
   3. You get an email notification as soon as the client-server receives an email and your device is connected to the internet, irrespective of whether the mail is in your device's Inbox or not

The pic below is self-explanatory

Notifications or Notifications Alerts?

1. Help for Notification Alerts --> No Notification Alerts - Apple Community
2. To turn ON notifications selectively for apps, go to Settings > Notifications > Scroll Down to the App and turn ON notifications... Change notification settings on iPhone - Apple Support (IN)
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3. Some apps have their own notification settings over and above the iOS notification settings: An example of Instagram Notification settings is given below
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Customise your Mail notifications on your iPhone or iPad