“ | Hey handsome, wanna dunk the clown? | „ |
~ Her one line.[src] |
The unnamed woman was the birth mother of Abe Simpson's son and Homer's half-brother Herb Powell. In the script, the character is named as "carnival floozy" and the night she and Abe meet was 1951.
History[]

A picture of the carnival woman addressed to "Abe-e-baby" in The Simpsons Uncensored Family Album.
When Abe visited the local carnival at Springfield (while courting Homer's mother), he was drawn to the young female worker at first sight. She had sex with Abe for money, and a year later when the carnival returned to town, Abe discovered that she had given birth to their son and they decided to leave him at the Shelbyville Orphanage. He was adopted and named Herbert Powell who went to Harvard and founded his own auto company Powell Motors until he was went bankrupt and lost everything.
Her photo in the Simpsons family album with a caption written by Marge, "I found this in an old Almanac that Grampa left in the bathroom. I don't know what her name was, but I do know that Homer's once wealthy half-brother, Herb, was the result of their short-lived affair." ("The Simpsons Uncensored Family Album")
Herb Powell was "conceived behind a Dunk–the–Clown booth of a traveling carnival" according to the bio on the back of the trading card.