While this guide is designed to help you choose a scene at the appropriate time, you could run a Shortcut as an action from the notification, or even call your automation server.
Step 1: Create a new Pushcut notification
Add a new notification in the Pushcut app.
Set a name and enter title and message.
Step 2: Select a HomeKit scene as action
Tap Add Action and select HomeKit.
Enter an action name and choose the scene you want to activate.
Tap Done, Save. Use Copy URL to copy the webhook URL to the pasteboard.
Step 3: Add a HomeKit automation
In the Home app, add a new automation (requires a HomeKit Hub).
This example uses Time Of Day > Sunset, When I am home.
Step 4: Paste webhook URL
Select Convert to Shortcut instead of a scene (under Advanced at the bottom).
Add the URL action and paste in the webhook URL of your notification.
Add the Get contents of URL action and make sure the URL parameter is set.
Tap Next and Done.
Step 5: Control your home!
That's it, you are all set!
It is easy to add multiple scenes, or select different triggers.
If you switch your URL Method to POST in HomeKit, you can specify dynamic texts, titles, even actions as [JSON body/support/notifications#web_api).
Turn on Keep Notification on your HomeKit actions to execute multiple scenes from one notification.