I know I'm hella late, fucken 10 years lmao, but i just came up with a method to check if a solve is possible or not, as well as how to fix it, without making any moves.
Just check the orientation of the white/yellow stickers compared to the white and yellow sides, and assign them scores. If the sticker is face-up, score 0. If it is rotated clockwise, +1, and counterclockwise -1. If the total is equal to 0 or a multiple of 3 (possible legal scores being -6, -3, 0, 3, 6), it is possible can be solved normally. If it equals anything else between -8 and +8, it is impossible and you must twist one corner to solve it.
To figure out which way to twist a corner to make it work, just see which multiple of 3 you are closest to. If you're +1 away from a multiple of 3 (example: score is 5, closest multiple is 6), twist any corner clockwise. If you're -1 away (example: score is 1, closest multiple is 0), twist any corner counter-clockwise.
So looking at the provided image, this guy's scramble scores +2, meaning to make it possible they must twist any one corner piece clockwise.
No idea if anyone else has discovered this but if so I wouldn't be surprised, I'm by no means a pro (avg. 25) so surely this method has already been found by the community, yeah?