Anna Christie. When Anna was a small child, Chris was constantly away at sea. Then when her mother died when Anna was five, Chris sent her off to live with maternal relatives, who abused her and raped her until she escapes, after which Anna resorts to prostitution. Chris claims he did it to protect her from the sea, but Anna blames him for all the suffering she's gone through.
Brooklyn. The titular character's parents are split up before she's born when her father is called away to war; her mother commits suicide a few years later, and Brooklyn is raised in a convent. After she grows up, she travels from France, where she's lived her whole life, to America, in search of her father. Who doesn't want to be found, and gets really upset when she shows up.
The Baker from Into the Woods was abandoned by his father, the Mysterious Man, after his mother's death. He in turn almost abandons his child.
Seymour Krelborn from Little Shop of Horrors was adopted by Mr. Mushnik, and he just makes the poor guy sweep the flower shop and sleep in the basement.
In West Side Story, near the beginning it is revealed Riff lives with Tony and his parents for unknown reasons.
The musical Wicked. Elphaba's mother dies while giving birth to Elphaba's sister. Her dad loves her sister, but is cold and distant towards her because of her looks. He dies later as well. However, Dad is not her biological dad at all, which Dad may have known and may be part of the reason he didn't like her. Her biological dad is the Wizard. She doesn't find this out and he doesn't find out until after her presumed death.
Georg Büchner's play Woyzeck involves the titular Anti-Hero killing his girlfriend and their child being abandoned by the neighbours. There's also a scene of an old woman telling the following story to her grandchildren:
"Once there was a poor child with no father and no mother. Everything was dead and there wasn't a soul left on Earth. Everything was dead and the child went out and searched day and night. But since there was no one left on Earth he wanted to go up to Heaven, and indeed the Moon looked down kindly at him, but when he got up to the moon it was just a piece of rotten wood. So he set off for the Sun, and when he got there it was only a withered sunflower, and when he got to the stars they were only golden gnats that a shrike had stuck to the blackthorn bush, and when the child wanted to go back down to Earth, it was just an upside-down chamber pot and the child was all alone. Then he sat down and cried and he's still sitting there to this day, all alone."
Shrek The Musical has this by adaptation: Ogres are kicked out of the house at age seven to go and find their own lair. This is further compounded by the fact that ogres face Fantastic Racism and Shrek was in fact chased and nearly killed by a mob within the year. Given Shrek's determination to write his own story by the end, it's unlikely he'll follow the same script.