To give them raw oils, I'd either add it to their feed or feed them treats, stuff like wholemeal bread with cold pressed extra virgin olive oil in it; generally I'd use the bread as a carrier for whatever else I was giving them like finicky powdery things that otherwise wouldn't mix well with feed, or for things I only feed them once in a while, like the once-a-month worming. So in that case I'd mix it with cayenne or similar, but in the case of a prolapse I wouldn't feed her that.
To give some oils, you could put it in something so they can drink it, or add a bit to feed, or use bread, or whatever you like. I guess if you put it out in a pan first and see how much they drink it would give you an idea of how much they want/need, but it might also act as a laxative if they have too much. Having said that, a laxative is a good thing once in a while, like a fast or cleanse or detox. I doubt they'd continue drinking it to excess day after day.
If possible, you should watch her and see when she poops... Does it come out alongside the prolapsed bit, or does it come directly out of it? That will answer which sort of prolapse it is... Hopefully. It might not answer anything.
The fact that you got an egg out of her makes me wonder if this is an intestinal prolapse rather than the usual sort. I don't know but I've not heard of a prolapsed hen laying. With sheep, if there's an intestinal/bowel prolapse they literally cut off everything that sticks out and sew the edges back onto the anus. But some folks use a pad and harness combo, especially with a uterine prolapse, and push back in whatever is hanging out, and hold it there. Some staple or sew the relevant orifice mostly closed. Not sure any of this would help with a bird.
An old treatment for prolapse in poultry was this: Massage the prolapse with grease, carefully press it back into position, and place a pad of cotton wool soaked in witch hazel extract over it, then lightly bind to the body. (I know, easier said than done, and getting pure witch hazel anythings is not so easy sometimes.. And the adulterated stuff could do damage). Anyway the treatment would of necessity involve fasting for a day and then the next day feeding a very light diet of mainly laxative and toning things like greens, bran mash, molasses and buttermilk. Nothing that encourages normal strength bowel movements, so no serious solids or normal feed. During this treatment stop feeding everything that encourages laying (so no seaweed or layer feed). I don't know what sort of grease they meant. Personally I'd probably use a little cold pressed olive oil. If it's not cold pressed it's already 'dead' (as in its active properties are cooked into oblivion) and would go rancid quickly.
Best wishes and I hope she recovers.
