It didn't mentioned about whether records his voice or someone else. So answer is vague.
A man who records voices
~ 오디오 엔지니어
"The" man recording his voice. <= Not accepted.
"A" man recording his voice. <= Suggested.
As far as specifics, a man is correct. As far as usual korean speaking, yes it doesn't matter whether it's a man or the man
the man who records his voice - accepted 17Oct 2018
A man records his voice- not accepted
"A man records his voice." is a full sentence with Subject + verb + Object
= 남자가 (자신의) 목소리를 녹음합니다.
"A man recording his voice" or "A man who records his voice" is a phrase with a Noun + relative clause (acting as a determiner)
= (자신의) 목소리를 녹음하는 남자