I'm all coughs coming after your teacher
Continues with the last line about even going through rain, snow or fevers with no breaks.
He’s saying that he’s sick and trying to deliver (a double meaning like the mail man deliver and delivering bars) to your teacher. He’s probably saying hes' better than your favorite rapper because in a sense they are your teachers/inspiration if you ever pursue rapping, or just a teacher in a traditional sense.
“I’m all coughs” meaning he’s “sick” as in he’s nice, he’s ill, he got skills, etc. and he’s “coming after your teacher” meaning he’s out to take down your favorite rapper.
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