Android: Launch “Chrome” app. Open the website or web page you want to pin to your home screen. Tap the menu icon (3 dots in upper right-hand corner) and tap Add to homescreen. You'll be able to enter a name for the shortcut and then Chrome will add it to your home screen.
iOS: Launch Safari on your iPhone or iPad. Navigate to the site that you want to create a Home screen shortcut to. Tap the Share icon (the square with an arrow pointing out of it) at the bottom of the screen. Scroll down to the list of actions and tap Add to Home Screen.
On the settings page you are able to enable push messages. Take the following in account: Push messages from this site work the same in modern browsers on phones as on laptops/desktops (Safari on OSX (Mac) and iOS (iPhone) excluded). Also in OSX are messages disabled for all browsers by default. This can be enabled in system preferences -> notifications.
In the browser settings push messages can be disabled per website