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If one speaks a foreign language making bad mistakes so that his speech sounds strange and broken, then he is speaking the word I'm looking for English (Spanish, Russian, etc.). Is there such an expression in English?
In Russian we say "ломаный русский (английский)", which is literally translated as "broken Russian (English)".
Yes, there's the same term in English, "broken English". First of all, there's a massive difference between having an accent and speaking poor English. You can have an accent and be very good at speaking a language. Think of former US Secretary of State Henry Kissinger, who always kept his heavy German accent, but was a diplomat and negotiator, both things that require you to have good command of the language you use. There are many people like this. You can describe an accent as "heavy" or "strong" if it's pronounced. Otherwise you can say a "mild" or "slight" accent. People who speak with even the heaviest accents, even to the point of being incomprehensible to an individual, may speak excellently; this would not be broken English.
broken English
phrase
if someone speaks in broken English etc, they speak slowly and make a lot of mistakes because they do not know the language very well
Macmillan Dictionarybroken
adj.
5.c. Spoken with gaps and errors: broken English.
American Heritage Dictionarybroken
adjective (INTERRUPTED)
C1 interrupted or not continuous:
He tried to explain what had happened in broken English (= not spoken easily and stopping a lot)
Cambridge Dictionarybroken
adj.
b : imperfectly spoken or written
broken English
Merriam-Webster Dictionarybroken
4. adjective [ADJECTIVE noun]
If someone talks in broken English, for example, or in broken French, they speak slowly and make a lot of mistakes because they do not know the language very well.
Eric could only respond in broken English.
Collins English Dictionary
Broken English Wikipedia article
Some definitions mention speaking slowly, or speaking with gaps, or making mistakes. However the term could generally be taken simply to mean that you don't speak the language well. In other words although the "broken" in "broken English" may suggest the "breaking/separating" of sentences with gaps due to difficulty in finding the right words, this is not purely essential. It may just mean you make a lot of errors. Also note that "broken" can be used for other languages. You can see the example above of "broken French", though this is less common.
I noticed your question in a comment:
So, would it be correct to say, that "broken English is the most widespread language in the world"?
Broken English isn't a language in the same way as French, Arabic or even Sicilian (which is called a dialect in many cases). But if we look at some definitions of language we see that it actually might make sense:
language
6.a particular manner or style of verbal expression:
your language is disgusting.
Collins English Dictionary2a : form or manner of verbal expression specifically : style
the beauty of Shakespeare's language
Merriam-Webster Dictionary5.A characteristic style of speech or writing:
Shakespearean language.
American Heritage Dictionary
The first example, "your language is disgusting", is obviously not referring to a language such as French or English, but the manner of speaking or certain words used.
In the second two examples we have "form or manner", or "style", for example "the beauty of Shakespeare's language" and "Shakespearean language".
The phrase "the beauty of Shakespeare's language" is not referring to Early Modern English or Elizabethan English, it's referring to the style of Shakespeare's language.
In light of these meanings I don't see the problem with saying what you want to say. It would probably be taken humorously, because broken English isn't a language in the way most people understand it, that is, like French. If you feel comfortable being facetious and irreverent, that's totally fine to say. It's quite funny. However if you're writing seriously, maybe for a school paper, you should probably avoid it and rephrase it.
Broken English wikipedia
... is poorly spoken or ill-written versions of the English language.
Many times called pidgins, there are a number of well known variants. Pidgins are a simple form of a commonly spoken language that's used by people who aren't fluent, for communicating basic information. In literary use, these can be used to display 'foreignness' of characters. Alas, I know of no idiom to describe this.
If it's spoken with a thick accent, you can use the adjective accented:
Language or speech that is accented is spoken with a particular accent.
I spoke good, but heavily accented English.
Collins
If a person is making many grammatical mistakes, the term for this is broken English:
(of a language) spoken falteringly and with many mistakes, as by a foreigner.
a young man talking in broken Italian
Oxford Dictionaries
Note that, as the first example implies, someone whose speech is heavily accented may not have any troubles with grammar. Likewise, it's possible that someone could speak without an accent, yet make many grammatical mistakes (this happens mostly in fiction).
