Hi, I've told Anna that ken is a good guy but i had refused his offer of doing my groceries. Is it possible to trigger the dialogue option "groceries one and the invitation to my house" by manipulating my save file or i am stuck with Ken that can't do nothing?

I'm stuck on the same situation lol, Like the mission with mario

If you haven't agreed to take the photo with Anna, Ken's quest ends then and there as far as I'm aware.

If you are asking how to take the photo, it's on Thursday -> Hallway -> Garage -> Entrance and there should be Ken and Anna.

I haven't ever refused Ken buying groceries so I can't help you with what happens if you decline.

I don't think you can manipulate save files easily. I just restart if I've completely screwed something

I don't understand why you would refuse Ken's offer: it's literally saving you 250 dollars and a trip to the Store in the Mall every week...