Hi Ludomir,
Probably your best option starts with a webhook. Webhooks fire when server-side analysis processing is complete and send the Quality Gate results as the payload. From there, an automation you build on your side would kick in to either retrieve the project PDF report or to build a report from the web API.
The best way to master the API is to perform the desired action via the UI and eavesdrop to see which calls the UI made to accomplish the action.
You may also find this guide helpful.
HTH,
Ann