Should I include the story of the dish in the photo?

Yes, if a detail of the table or the pass explains the dish without a paragraph. No, if the story only fits in a caption and the photo becomes a still life that is not your room. Pick one fact the floor already says out loud: the oven, the market, the cut. The miss is building an origin tale with props the guest never finds when they sit down. If the detail is included, it sits in a frame beside the plate, not instead of the plate.