I've just been having a hard time finding these stories. I've seen Usagi Drop, Celeb to Binbo Taro, and Zenkai Girl; all of which feature single people who are raising children on their own. Many drama summaries don't outright tell you the main lead is a single father or guardian so it's been very difficult to find any.

Sometimes it doesn't even have to be a father or a gurdian. Just somebody raising a child is fine as well. For example, in the manga and anime Aishiteruze Baby, a high school student learns responsibility through raising his niece.
I'll recommend you some you just have to look if it's fit your category...Some of those I will recommend were dramas that the main character was raised by a single parents. I'll just put an * to them for you to know... Bara no nai Hanaya Thank You - Korean Daisuki!! Stars Falling From the Sky - korean *Engine - It's not mainly what your asking for but the main lead is taking care of children in orphanage Good Life ~Arigatou, Papa. Sayonara~ *Powerful Opponents - One of the main guy was raising a daugther *Ouran High School Host Club - the main character was raised by the father Oh! My Lady My Girl - Japan Marumo no Okite MoMo Love - the main character we can say that she is basically raised by her 4 brothers (taiwan) Mother - Japan Speedy Scandal - korean movie Baby & I - Korean movie
Korean long drama (50 ep) My Precious You. Single dad raising two kids after divorce.
Namida wo fuite (J-drama) Nobody knows (J-movie) maybe Rinne no Ame (short special) in the last two, older brother take care of his younger siblings.
The only one I remember right now is My Girl (Japan)
Prime Minister & I, korean drama
baby and i momo love as darkangel said..... haeundae lovers she isn't a child but she raised by her 3 uncles and it's like a child and special person for them cuz her father
ohh I forgot, Miss Korea :)
Bara no nai Hanaya
Relationship btw father and daughter....mixed of family,mystery and romance...

Tonbi
this is a very touching and heartwarming drama about relationship btw father and son
Cheer Up Mr Kim! About a single dad who raises three adopted children on his own!
Movies:
Girl - a movie from Japan that features a single mother trying to raise her son. She is looked down upon by her peers because they believe her son isn't getting the correct "role model" as he's growing up without a father-figure around.
Himizu - A son who lives with his more often than not drunk mother. His father comes home occasionally to bum money off the mom and throw his fists around a bit. Shows the psychological damage this has on the son.

Dramas:
Down With Love - a Taiwanese drama about 2 children who've lost their parents and are taken in by their uncle. Their uncle doesn't really know how nor want to deal with children, though, so he hires a nanny to take care of them. The relationship between the uncle and children gradually becomes warmer and more family-like
Heaven's Mandate - a Korean saeguk featuring a single father and his sickly daughter. The father is a physician in the Royal Court, from which he gets the medicines to treat his daughter. They're separated for some time, however, and a lot of the story is devoted to the father finding a way to get back to his daughter.
Brother Beat - a Japanese drama about a single mother and her three grown sons. The mother has an unconventional approach to raising her sons. Mostly a light and fluffy drama
Mother - a Japanese drama about an abused daughter and her teacher who decides to raise the girl herself and take her out of that situation. Also focuses on the teacher's relationship with her mother, who also had to raise her daughter on her own.
My Prince - the Mainland Chinese version of Brother Beat, about a single mother and her four grown sons.
Sad one: Korean Drama "Golden Rainbow" Funny one: Korean movie "Baby & I "
let's go to school sang-doo (korean)