Its not mentioned in the movie but if you read the excellent graphic novel Flash Gordon Zeitgeist the writers actually bring back Klytus and do try and explain away dome of the mystery. This graphic novel tells the entire original story of Flash vs Ming but encompasses not only the original strip but the serial, the animated series and the movie as well so elements from all of these are fused together. As a result we get Flash's life being set around pre-World War II, we get Klytus in the story plus the yellow robots from the animated story amongst other things. In Zeitgeist Klytus is a scientist whose face was ruined in one of his experiments. Highly recommended.
Another thing I was reading the other day, according to Peter Wyngarde at the 35th Anniversary Celebrations he was told by the producer that Klytus would have been in Flash Gordon 2. It was supposed to be a rejuvenated Klytus who picks up Ming's Power Ring at the end of the first movie. The sequel would have featured Klytus in a different mask and portions of Ming's face would slowly have formed on it before becoming whole by the end of the movie.