The phrase "last scrap" means the final bit or remnant of something. It refers to the very last piece or small amount remaining of a particular thing. Full definition
He leads a Church that long ago let itself become a corrupt parody, yet he is a last scrap of the old succession, with the fisherman's ring upon his finger. (firstthings.com)
Last night that meant cleaning the last scraps from the carcass of a chicken Cathryn had roasted earlier in the week and making a chicken salad. (forums.egullet.org)
I did munch on the last scraps of my Chocolate Peanut Butter Popcorn Balls / Bars. (sinfulnutrition.com)