You can see the plugin documentation here: Splashscreen - Apache Cordova
You need put this code into your push event into your app:
When using cordova-plugin-splashscreen you can configure it to only appear the first time you open the app with this preference
<preference name="SplashShowOnlyFirstTime" value="true" />
The first time means that if the app is in background and you open it (bring it to foreground from the icon or from a push notification), the splash image won't be shown again, but if you fully close the app, then it will be displayed again when you open the app.
Also, on iOS there is a thing called "Launch Images" that look exactly the same as the Splash Images, and you can't get rid of them.