Is this also used for someone, who is a trainee or apprentice for several years, in order to master a skilled trade, seeing that הִתְמַחוּת means literaly specialisation (הִתְמַחָה: to be recognized as an expert, become a specialist)? But there is also the term חֲנִיכוּת apprenticeship.
I actually hear התמחות (in relation to profession) only in two uses: medicine and law. In medicine that's the stage where the doctors have already received their MD, and are working in a particular ward for several years until they graduate from that and are titled specialists in the field. I think in law it's similar; and maybe it's similar in civilian engineering, accounting, architecture, but I just don't hear about it. So I guess that's what you're talking about, I don't even know what's the English term for that.
In the world of high-tech (where I happen to work) we have interns - that's typically students who work at the commercial company in parallel to studying, or during school vacations, typically for months rather than years. The term we use, as part of English terms infiltrating our whole professional talk, is אינטרן and אינטרנים and אינטרנשיפ... Or maybe it's just the parts of high tech I've been to.
For "apprentice" we have the words שוליה and מתלמד. The former is not a thing in modern day Israel, but would be used in talking about other times and places. Maybe there are people who de-facto work as apprentices, and I guess they may be referred to as מתלמדים, but it's not a formal distinction.
The word חניך is used only for learners in courses outside Academy and regular schools, and provided they are young (for 30 year olds who go to a professional course, I guess חניכים would not sound appropriate). In particular, it's used for children who go to after-school lessons of sports, music etc, in the context of the lesson. It's also used for members of youth movements.