Woobie of the Week

So we have Mystery of the Week, where the heroes solve a new mystery ever week. And we have Monster of the Week, when the heroes fight a new bad guy every week. We even have Patient of the Week, when the heroes are doctors and they help out a new sick person every week.

And then there's this guy. He's not dead or in danger of dying-- he's just troubled in some way. Maybe he's angry at God because his wife died, or he's in debt and needs help, or [insert touching story here]. Whatever the problem, our main character has to figure out how to help him. And then the Woobie promptly disappears, and next week our hero finds someone like him all over again...

These series tend toward being spiritual and glurgey, but really it's up to the skill of the writer.

Anime and Manga

 * XXX Holic: The heroes go about, helping addicts of various types.
 * In Ballad of a Shinigami, Momo the shinigami finds some way to relieve the pain of the dead or the ones they are leaving behind each week.
 * Tokyo Babylon has A LOT of these.
 * Princess Tutu is mostly this sort of thing in the beginning, with the heroine helping people solve their issues through dancing. It's Better Than It Sounds.
 * Basically, every other One-Shot Character in the Sailor Moon anime is one. It get annoying after a while, as the Senshi are perfectly fine helping out the character they barely know, but they spend no time with their actual friends. Remember Naru? Usagi probably does not.
 * Mushishi combines this with Mystery of the Week. Most episodes feature people in incredibly unfortunate circumstances due to the effects of various mushi. Then Ginko arrives, figures out what kind of mushi he's dealing with and sees if he can help. Sometimes, he can't.
 * Hibiki no Mahou has these, one of whom is Hibiki herself.

Live Action TV

 * Highway to Heaven
 * Touched By an Angel
 * The similarly-premised Twice in a Lifetime, where an angel allows a one-shot character to revisit a moment in his life when things started going wrong.
 * Wonderfalls: A bunch of animal figurines that may or may not be God annoy heroine Jaye into helping strangers on a weekly basis.
 * Joan of Arcadia: Joan helps a new stranger every week at the suggestion of God himself.
 * Ghost Whisperer: "Distressed ghost who refuses to pass on" of the week.
 * On Fantasy Island, each week a different group of guests come to the island and learn some sort of life lesson
 * The Love Boat likewise has a new set of guests every week, all of them looking for love on the cruise ship.
 * Quantum Leap has this built into its premise: Bakula jumps into a new body every episode, and that person, or someone around him, is always in distress. The series catchphrase, his mission, is to Set Right What Once Went Wrong; once that's accomplished, he leaps into the next body. It's heavily implied to be the result of divine intervention.
 * 21 Jump Street, the episode where the cops go undercover and there's this chick who's all introverted but seems to know about the crimes that are going on, and she's all disturbed. Turns out she's psychic.
 * Season 4, an episode called "Haunts in a New Age"... A psychic teen says she can predict when an arsonist will next strike during a Halloween dance.
 * Kamen Rider Fourze has an interesting twist on this, combining Monster, Victim, and Woobie of the Week into a single package. The MOTW is created by a human "Switcher" using a device called an Astro Switch; the Switcher is someone with a grudge who's so hell-bent on revenge that they're blinded to the fact that the Switch will eventually kill them. So instead of just beating up the MOTW, Fourze and his team reach out and try to befriend the Switchers so that they know there's somebody who cares.