I have created a new plan in Planner under "My Plans". So this plan is private and not in a group. Is it possible to access it within the Power Automate flow? Seems like you need to have a Group Id to access plans in planner, so is it even possible for private plans?
Use the classic designer. Then open your plan and look at the URL. In it you will see groupId as part of the URL:
Copy everything after the equal sign (=) up to &planId. Then choose Enter custom value for the Group Id and enter the value. If you copied it correctly, when you click on the Plan Id drop-down, you will see your private plans listed.