I think you can allow take backs in unrated games but never looked for it or used so dont quote me.
Forgiveness mode

Should be a mode where takeback offers can be made between consenting adults, which would nullify any rating change.

Only take-back is under Play, Daily Chess. There is a Rated on/off control. I can't tell you where because -- beware -- if you initiate a game and there are no players available, the system keeps looking and looking... forever. Not that it matters except I'd love to provide a compete answer. Evidently there's not much demand. Thx all.
Is there a mode where either live player can offer a move do-over? Sometimes my opponent will play well but blunder -- of course I <cough> never do. Would be a good learner's feature to offer undo move so game might continue competitively. Thx.