1. Childhood (This writing is for speaking test)

When I was little young [You are still young at 24 by today's standards] , I used to polish my parents’ shoes to earn pocket-money. And When I had the time, I helped my mother wash the dishes too as well. Whenever I did it, For this help, My mother she would give me a Korean princely sum of a thousand Won (about US$1) used to give me 1,000Won,which is Korea money.

I saved the money. by Little by little, and finally I was able to buy bought a robot what I wanted. Now I clean up the house for free!!!

Since My parents already hand spent a lot of money for on me.  So I feel guilty.
When I get a job in the future, they will get a lot of spending money from me.

In return, [This sounds like a transaction and not out of love] I’d like to give them spending money.

As I correct your entries, I start to get to know you a little better.  We share a common tradition and similar culture.  Glad to know you.

1. Childhood (This writing is for speaking test)

When I was young, I used to polish my parents’ shoes to earn pocket-money. And When I had time, I helped my mother wash the dishes as well. Whenever I did that, My mother used to give me 1,000 Won, which is Korean money.

I saved money by little by little, and finally I bought a robot what that I wanted. Now I clean up (this isn't necessarily wrong, you could say "Now I clean up around the house for free".  It just has a more common sound.) the house for free since my parents already spent a lot of money for me. so I feel guilty, so when I get a job in the future, in return I’d like to give them spending money.

Well done.  My corrections are just to show you what someone would more naturally say in a given situation.