Fable Anniversary The Sick Child is one of the side quests that you can get after completing the Prologue. The quest is available in Bowerstone South as soon as you visit the town.
You can get the quest from a little girl in Bowerstone that leads you to her mother. She explains that her son is sick and the only one who can cure him is a witch from Bowerstone Quay.
But in order to make an antidote she requires four Blue Mushrooms.
The Sick Child is a lengthy side quest that can only be completed around the second chapter of the game.
You can find more information on how to complete the quest in the next Fable Anniversary walkthrough.
When you first enter Bowerstone you will notice that a small girl is following you around. Speak to her to start the quest. Follow the girl to her mother and then speak to her to learn about the task.
After that go to Bowerstone Quay to find a witch that can make an antidote for the sick child. From this point your task is to find four special mushrooms which you can get only from certain regions.
For the first Blue Mushroom return to the Picnic Area where you battled the Queen Wasp in Wasp Menace main quest.
Here you can find a woman that asks you to make her laugh. Simply choose an expression that will make her laugh and repeat the process a few times until she gives you the mushroom. When you get back to Lookout Point you can find two men that are definitely on mushrooms.
Listen to them to learn a clue about the second Blue Mushroom. You find out that is in the Heroes Guild area. When you complete the Fishing Lessons side quest you can go to Heroes Guild.
Head to the Demon Door and fish the mushroom from the pond. While at the Guild you can start the quest for another Blue Mushroom. Go the Guild Woods and talk to the man that is there.
He asks you to deliver a letter to a girl in Oakvale, thus you wont be able to get the mushroom until you unlock that area (unless you kill the man on the spot and take the mushroom from him). Latter in the game you will reach an area named Barrow Fields. Here you can find a trader that is willing to sell you your third Blue Mushroom for 1500 Gold (or you can steel it if that suits your play style).
When you reach Oakvale go to the east side of the town to find the girl by the well. Give her the letter and tell her who sent it (or lie and say that is yours).
After that you can return to Guild Woods to collect your fourth Blue Mushroom. Next, go to Bowerstone Quay and give the mushrooms to the witch in exchange for a cure.
Return to the woman in Bowerstone South to collect your reward and complete the quest.