Can you omit words from a sentence and correctly cite that text?

Traditionally, the sign for elided words is the ellipsis, which is three dots. Those used to be created with periods. In modern word processing, it is its own symbol.

Thus,

Trafficking in children from Togo, Nigeria, Mali, to Cote d’Ivoire’s plantation and domestic servants in Gabon, and of women from Ghana, Nigeria, Mali, and Sierra Leone as exploited sex workers in countries of the European Union has also taken root.

becomes

Trafficking... domestic servants... [from] Sierra Leone as exploited sex workers in countries of the European Union has also taken root.

The brackets are necessary around the "from" because it does not appear in that position in the original text.

There are a couple of cautions. Some of the other answers here seem to suggest that brackets are necessary around an ellipsis that's not in the original text. That's not a rule I've ever encountered, but it might be the new academic standard. Your own discipline may have a style guide that covers this situation.

Also, it's considered bad form to chop up someone else's words too much, since you can dramatically change meanings that way. Therefore, it may indeed be better in this case to just paraphrase the original, and indicate that fact when you cite it, as in the accepted answer.

You can use the ellipsis sign ... to indicate the part of the sentence you don't want to include in you citation.

The cleanest quoting approach would be as follows:

"[Trafficking] domestic servants in [Sierra Leone] as exploited sex workers [...] has also taken root"

Brackets can be used to show that you've summarized or edited excluded text, not merely removed it. But in this case I'd lean towards paraphrasing, e.g.:

Whomever (Whomever et al, 2017) make the case that domestic servants from Sierra Leone are increasingly being trafficked and exploited as sex workers.

If you think it's essential you can add the full quote as a footnote. This allows you to focus on the information you need (and avoid some of the given quote's grammatical issues) without sacrificing readability or altering the meaning of the original text.

Of course not; not as in your example.

You either cite the exact text, or you show how you've changed it. Is that much truly not obvious?

That is more, not less important when - as here - the original text is unclear.

"Trafficking in children from (anywhere) in countries of the European Union…" is broadly comprehensible only from assumed context, not from the text itself.

("… I wrote actually I only need it…" means what, please?)

Of course. You see this often, when people are citing a speech or another, drawn out text. What you'd see is a normal cite, but in every place something is missing, there's a "[...]" to indicate that you omitted something. In this case, yours would end up being

"Trafficking [...] domestic servants [...] from [...] Sierra Leone as exploited sex workers in countries of the European Union has also taken root"

Which is not pretty, and you might want to change this, but this is correct and you'd see it in a lot of academic texts. You still have to be careful not to alter the sentence too much or leave too much out of it, since that drastically decreases credibility.

Yes. You may be putting the information from that sentence into your own words, but you've still taken that information from another source, and you need to cite that source just like any other. Otherwise, as far as anyone knows, "Trafficking domestic servants from Sierra Leone as exploited sex workers in countries of the European Union has also taken root" is something you just made up.

Source: I wrote a dissertation and multiple essays at university and did a lot of paraphrasing, but I still had to cite the things I'd paraphrased.