Ok, first of all, if you are going to bring back an ST:TOS character, do you really want it to be Harry Mudd? Khan made sense.He was cool. But Mudd was a goofball. Going back to the Mirror Universe (a few too many times , probably) was cool. HFM can be forgotten. At least Rain Wilson plays him well, even if he isn’t really worth playing well in the first place.
Now for the time travel. Mudd seems to be able to rewind time at will in order to e enventually get the results he wants (the secret to operating the spore drive). This ability seems to be taken for granted, and only roughly explained in terms of special crystals towards the end of the episode.
We don’t know much about time travel, but what we do know suggests we would need immense energies to manipulate space and time. Perhaps the engines of a starship would be good enough, but we seem to be told that Mudd has no access to these. So, problem 1 is that there just isn’t enough energy.
But a worse problem may be the ability to manipulate time at all. Most physicists agree that time travel to the past would only be possibly in ways that would not change the known flow of history. That is, you could send yourself back to the time of Julius Ceasar if you had the right amount of energy and “machinery” , but you couldn’t push the dagger from Brutus’ hand and prevent the assassination. We believe so because if time manipulation in this manner were possible, it would allow logical impossibilities such as going back in time and killing any ancestor of yours, thus preventing your birth and cutting off the ability to go go back into the past!
So, Harry, you don’t get infinite chances to go back in time and neither does anybody else. Sorry!
If you like reading about these ideas, check out Steve Bloom’s book,
The Physics and Astronomy of Science Fiction available both through McFarland Publishing and your favorite internet bookseller such as Amazon.