I'm trying to express next thing: I translated some text into another language for training/study that language, got some mistakes, these mistakes were shown and explained to me and now I'm trying to make up some new phrases which include the same weak points (particular grammar construction, word etc.) where I made mistakes previous time. So is "Work on the bugs" suited for this situation?
"Bugs" meaning mistakes or errors is a term used mainly in the software industry. It isn't normally used for mistakes or errors made except when writing software. You might occasionally hear it used in other contexts, but that is usually a case of someone wanting compare the other context to programming or a usage by someone in the software industry for whom the word is simply more common than alternatives.
A more mainstream way to talk about improving a text would be "editing the text" or (less formally) just "fixing the mistakes".
For other languages that might be difficult. I have had the pleasure to speak (online face to face) with a lot of others in numerous lands, and our single words and phrases can mean quite dirty facts when "just" said with no fast explanation.
I disagree with only the software aspect. "Working out the bugs" is used daily in all situations of reality here in the states. We work out the bugs to fix a problem, etc. Editing is simply making certain the words are organized or spelled correctly.