The most common and most likely answer to this crossword clue is the 4 letter word BUBO
I've seen this crossword clue in The Guardian
B
U
B
O
Buddy explains!
The straight part
We know that "that might come out of armpit" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case that might come out of armpit = bubo.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does BUBO mean?
BUBO noun- An inflamed swelling of a lymph node, especially in the armpit or the groin, due to an infection such as bubonic plague, gonorrhea, tuberculosis or syphilis.