Is it correct to begin with an adverb just like in: "A recently done research"...
That wouldn't be grammatical English, but not because of the adverb. "Research" in this sentence isn't a count noun, so "a research" should be something like "a piece of research" or "a paper" or something, depending on what you mean.
But that use of an adverb is fine aside from that. "A recently picked apple was waiting on the table when I got inside" is a fine sentence, as is "The recently elected president consulted with his staff."
I am hesitating over the word "done". If it's academic research, publication is usually the key date, and somehow "done" sounds a bit demeaning. Laundry is done; academic research is published, announced, pursued, concluded but not just done. So, if I were editing your text, I would write " Recently published research has been..."
But there is nothing wrong with starting that sentence with the adverb "recently" - why would anyone think so?
Study, not research.
And the best way is to just say "a recent study" (has shown that blah-blah-blah). The verb, or, rather, participle, is redundant there.
If you do need a verb, an appropriate one would be to conduct ("A study recently conducted at the University of Northern Unicorns has shown that blah-blah-blah").
A phrase like "he has done some research and found out that" doesn't necessarily assume an academic context. You could say to someone who complains that they don't like their newest phone, "Do your research before buying phones." Meaning, "study this subject matter", but not as an academic. Just google and find out things, read up on new phones and their features.
That thing where scientists in lab coats let rats run through mazes or attach electrodes to people's heads and then write down findings in lab journals, calculate statistics and then publish an article in a scientific journal—that thing is called a "study".