We have to identify the noun phrases, I've put brackets around them.

1. (the children) are playing in the back garden.
2. Justin came to school with his (younger sister).
3. (the old woman) hobbled into the kitchen.
4. Look at (the short ugly looking boy) who is throwing tantrums.
5. I've bought (that book about prehistoric creatures).
6. The villagers gathered by (the bridge over the stream).
7. The love watching (sunsets at Changi Point).
8. Jeremy hopes to work for his father once he graduates. Sorry i have no idea about this one, maybe for his father?
9. Augustine mentioned reading (this book in junior college).
10. The manager overheard his saff applying for (a job in a rival company).

Argh! Please help! I really don't understand this so well.

1. (the children) are playing in the back garden. YOU COULD ALSO INDICATE "IN THE BACK GARDEN."

2. Justin came to school with his (younger sister). YOU SHOULD INCLUDE "WITH HIS" SO THAT THE WHOLE NOUN PHRASE IS "WITH HIS YOUNGER SISTER."
3. (the old woman) hobbled into the kitchen. YOU COULD ALSO INDICATE "INTO THE KITCHEN."
4. Look at (the short ugly looking boy) who is throwing tantrums. CORRECT
5. I've bought (that book about prehistoric creatures). "THAT BOOK" AND "ABOUT PREHISTORIC CREATURES" ARE TWO SEPARATE NOUN PHRASES.
6. The villagers gathered by (the bridge over the stream). INCLUDE "BY"
7. The love watching (sunsets at Changi Point). OMIT "SUNSETS"
8. Jeremy hopes to work for his father once he graduates. Sorry i have no idea about this one, maybe for his father? YES, "FOR HIS FATHER" IS CORRECT.
9. Augustine mentioned reading (this book in junior college). AGAIN, THERE ARE TWO -- "THIS BOOK" AND "IN JUNIOR COLLEGE"
10. The manager overheard his staff applying for (a job in a rival company). AGAIN, THERE ARE TWO -- "FOR A JOB" AND "IN A RIVAL COMPANY."