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I want to add a perverted character to my story (don't ask why), a bit like Master Roshi from Dragon Ball or Minoru Mineta from My Hero Academia. However, I want to make them more sympathetic and give an actual reason as to why they are such perverts, not to justify their behavior, but to show them in a different, more tragic, light. You know, instead of just comedy relief.

Side note: just like in the examples I listed above, the pervert is punished almost instantly whenever they do something perverted. Nobody except other perverts encourage their behavior.

How could I do this? One idea I had would be to have them think they are unattractive, thus they're simply perverts because they know they have no chance with any of the girls anyway.

Two tropes you might want to look into are Chivalrous Pervert and Lovable Sex Maniac.

The Chivalrous Pervert would be thinking about sex a lot. He would be unable to avert his gaze when that would have been the polite thing to do. He'd be constantly making passes at women. There'd be no doubt in any woman's mind that he's very much interested. But there are boundaries he would not cross. He would be a gentleman to all the women he meets. He'd never make himself actually unpleasant to a woman. In fact, he'd be the first to jump to a woman's defence (even if she really doesn't need it, and can, in fact, defend herself and him).

The Lovable Sex Maniac is significantly more perverted, but in his case all the perversion is played for laughs. He's inappropriate (though, again, he doesn't cause actual harm). He's often over-the-top. He's so over-the-top, it's funny. And because it's funny, the reader is inclined to forgive the inappropriateness.

What those tropes have in common is that in both cases, the guy is actually harmless. You won't get sympathy for a character who victimises a woman by continually harassing her, for example.

There is of course the example of Humbert Humbert, from Lolita, but I wouldn't categorise him as "sympathetic". He is, however, captivating. This is achieved, I think, by him being so intelligent. His internal monologue is interesting to read. But he is captivating and repulsive at the same time, that's the beauty of that novel.

"Pervert" covers too much ground, or is too general a term. Pedophilia is a perversion, and you won't get any sympathy for a guy that wants to have sex with children under 10, or wants to see pictures of them naked, or masturbates in his car outside a grade school playground. Whether he actually harms anybody or not.

Now if you are just talking about a mild perversion, you can do it. Say having an orgasm while sucking on a women's toes. Or wanting to have sex while dressed in women's lingerie. Or any other perversion that nobody thinks may someday harm somebody.

So for the same reason, it would be difficult to make somebody likable if they fantasize about violent rape, and like that kind of pornography (simulated or not). Basically, they may "tilt" someday and actually harm somebody.

You will need to keep your character's perversion just weird, not potentially harmful.

Then you can make him sympathetic, for keeping a secret, for being embarrassed about it, for his inability to emotionally connect with another person sexually. Or, perhaps they can be happy for him, if he finds a partner that likes his perversion, e.g. a woman that enjoys whatever fixation he may have. There's someone for everyone, they say. (I doubt that is true, but you could write it that way.)

I'd echo the above answers that it's possible to write a likeable pervert provided they're harmless, and provided the scope is narrowed to that character and does not creep into the narration of the story. You don't want to alienate readers by allowing a misogynistic undertone to infect the narrative.

Think about why your character is the way he is. We struggle to feel sympathetic for someone who comes on to every woman he meets if he's cocky, arrogant and pig-headed (and we hate him all the more if he's charismatic enough to have at least some success with this approach), but we can feel sympathy for a man whose inability to interact with women as anything other than sex objects is an undesired symptom of a social interaction problem. We might even feel sympathy for a young man brought up in a sheltered household where women have been always been treated as objects, and we can come to like him once he gets into the real world, upsets some people, and must battle his own upbringing to find his place in the world. There's a story right there.

With enough skill and sensitivity, I think you could even make the reader sympathetic to someone with a violent or dangerous perversion. Your character might have violent or abusive fantasies, but he might be disgusted by these fantasies and he might feel that they are a kind of illness that he's desperate to be rid of. That's a very powerful angle to write from.

You can get some inspiration by remembering that today's norms are yesterday's perversions, and history is full of real life people who struggled to come to terms with what were at the time considered to be perversions. Brian Sewell once described his homosexuality as an "affliction" and a "disability" in an article which no longer seems to be online but is referenced on his wikipedia entry.

https://en.wikipedia.org/wiki/Brian_Sewell

Remember that sexual behaviour can be very compulsive and that can create terrible conflict in a person who is compelled to do a thing by their desires which they consider to be morally abhorrent. There's a lot of creative material there for someone with the determination and sensitivity to pull it off.

I'm sorry I don't know either of the examples you cite above, so if my answer is slightly missing the point, apologies.

My answer is an expansion of what you've touched on in your question.

Explain and justify why they behave the way that they do, that makes the reader feel sympathy for them.

This isn't even a fictional concept, but played out in real life over and over.

An abusive husband and father is that way because he was abused as a child by his parents.

A man who flashes women in the street does so because his mother always told him he was ugly and worthless and no woman would ever want to get to know him.

So show your character doing whatever it is they do, then think back to their formative experiences to try to work out why they feel they have to do that. Then include a scene that demonstrates this experience to your readers, showing the 'pervert' as the victim - this could be a flashback scene or the character could recount it to someone else.