What is the correct Chinese expression for "Only child"?
The Chinese expression for 'only Child' can be '独生子女(only child)','独子(only son)','独女(only daughter)'，or some more ancient or literary words like '独苗(only child)','单丁(only son)',单根独苗(only child or only descendant).
Native Chinese speakers may have other expressions like,

他膝下仅有一女。（He has only one daughter.）
他是那老汉唯一的儿子。（He is the only son of that old man.）
他是三代单传。（He is the only one son for several generations in the family.）

I hope the above helps.