Why did Harry Mudd get two episodes in the original Star Trek series, while Khan...

[TMC]

...only got one?

Because Harry Mudd is far more useful and practical character than Khan Noonien Singh was or ever could be.

Why?

1. The Augment storyline has always been problematic for the Star Trek franchise - When you strip away all of moralizing from the narrative, the core issue is that the Federation makes decisions for human beings (by outlawing genetic engineering/enhancement) that weren’t illegal in the 1960s when the story was first told, that wasn’t illegal in 1982 when the second film was made and still isn’t illegal. The fact that you can’t breed human beings with superior intellects and abilities is disturbing and a violation of the rights of Federation citizens.
2. Khan was a superman…and? - The story (Space Seed) was essentially self-contained. The Enterprise encounters frozen super humans, almost lose their vessel to them, defeats them and then maroons them on a distant planet. The film (also self-contained) simply showed what happened to those aliens, allowed them to escape and then ended with their being killed. The narrative begins and ends with the end of the story. Even the Enterprise story arc about Augments was self-contained, ending when the arc ended. Harry Mudd allowed for multiple tales to be told using a popular character. His stories were all open-ended.
3. Harry Mudd was a popular character - Technically, there four episodes that were supposed to be about Harry Mudd (the Cyrano Jones character in The Trouble With Tribbles was clearly based upon Mudd and may have been given to actor Roger Carmel had he not had numerous issues with drug abuse) Khan, while a popular character, was not viewed as being as popular at the time as Mudd was. The series knew enough to return a character who was popular with the fans.
4. Star Trek: TOS was cancelled after only three seasons - If the series had lasted longer, it is conceivable that Khan could have returned.. As it was with only three seasons and 66 individual shows, there simply wasn’t the time to revisit the Khan character until far later.
5. The original series was set in a different era - An amusing recurring character was a common trope in American television of the past. Many of today’s television viewers and filmgoers are jaded and they would quickly grow bored with an impish trickster. Khan played better to the somewhat jaded audiences of 1982 and then in the wake of the 2003 Battlestar Galactica reboot changing space sci fi almost completely, reusing Khan as opposed to Mudd seemed to be the better decision.

Harry Mudd was a more popular character in the past and that’s why he was used in several incarnations of the series. Khan became a more popular character after the 1982 film and he was used in the present.