AWS IoT Core uses MQTT protocol to communicate with your Raspberry.
You have SDKs written in almost every languages.
You can do D2C and C2D.
An easy solution for you would be , when the user clicks the button on your mobile app, it triggers a function on your API that publishes to a MQTT topic and on the raspberry you just need to subscribe to the same topic to receive the message.
You can find some nice tutorial directly on AWS documentation :
Hope it helps
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