I would suggest to use Knife Project for this task. Please be aware that this approach doesn't give you a clean topology.
You will need to add another object that is used to add the cuts. In your case I would use a Circle (Add > Circle in object mode). Select the Circle first and then the object you would like to apply the knife project on. Position your viewport perspective so that the Circle is placed where you'd want the cuts in the target mesh to be. Switch to edit mode (Tab), open the operator search (F3 or Edit > Operator Search), find the Knife Project operator and execute it.