Three years ago i stopped, I quit my professional carreer because iI needed to take care ofor my children.

1. When talking about leaving a job, the verb “quit” is most commonly used
2. Whenever you use the word “I” in English, it must be capitalized
3. In English, the phrase is normally said as “take care of ___”
There’s also a few spelling errors here

Now iI want to improve my eEnglish so that iI can apply for a job again.

The word “English” is a proper noun and must be capitalized

I hope that in one month iI wouldill be able to speak and whrite much better than iI do now.

1. You should use “will” here instead of “would” since you’re talking about the future
2. The word “white” refers to a color. I think you mean the word “write”

I that Hope iI can.

This phrase works in informal conversations when it’s already been established that the subject is you, but in this case, you should include the subject, that being you

Why lLearning English aAgain mMatters to mMe

Title capitalization rules

Three years ago i stopped, I left my professional carreer because iI needed to take care ofor my children.

We tend to use the term "leave" when ending a job.
You can say "to take care of" or "to care for"

Now iI want to improve my eEnglish so that iI can apply for (a job / work) again.

"English" and "I" are always capitalized.

"Apply for a job" highlights one singular job, like perhaps you have a specific position in mind, such as with your last employer. But if you want to speak generally about many jobs, "apply for work" is slightly more general

I hope that in one month iI wouldill be able to speak and whrite much better than iI do now.

We are talking about one month in the future, and not a hypothetical scenario, so "will" not "would"

HI hope iI can (do it).