Is there a universally recognized term for a sequence like the following?
It's not a verb phrase, but it doesn't seem to be a verb group in the sense of M. A. K. Halliday's framework, either.
What exactly is it called?
I'd appreciate your help.
Take care of, like make love to, have a drink with, make a date with, or millions of others, is a fixed phrase. That means it's always reproduced using the same words (and often the same intonation), and does not change. Of course, long frequent phrases often get shortened by habit, and the of at the end is most likely to turn into a schwa, like kinda, sorta, wanna. Often the meanings are changeable, especially in metaphoric or pejorative senses.
If a mobster tells a henchperson to "take care of him", that probably implies violence for him. But if a nurse is told the same thing, what's implied is at least official recognition of him, and possibly medical care for him. But they'll use "take", "care", and "of" in both cases.
This particular type of fixed phrase is a verb phrase onset that is turning into a transitive verb itself, so the object of of functions as the object of the transitive verb take care of, as shown by passivization, for instance:
