How to convert one unicode char to second looking exactly the same?

I will try to explain my problem. I am working with Czech special letters in MySQL database, such as ěščřžýáíé, etc. I was having problem when searching in database, because I was searching for exactly the word, which I knew was in the database, but no results were found.

Later I discovered, that somehow I have two "versions" of characters, which look exactly the same, but they are different in Unicode. Please take a look at example here. Both "á" and "á" look the same and should be treated as the same. So now I need a way how to convert one to other. Preferably using MySQL or PHP.

As you can see on above picture, the first character is actually made of two characters "a" and "´". The second characters is just "á" as single character and that is what I would like to convert the first one to.

Any ideas how to handle this ? Of course, I am talking about all the Czech characters, not only "á", I used it only for example.