The most common and most likely answer to this crossword clue is the 3 plus 4 plus 6 plus 2 letter word GOT ONES DANDER UP
I've seen this crossword clue in Financial Times
G
O
T
O
N
E
S
D
A
N
D
E
R
U
P
Buddy explains!
The straight part
We know that "became somewhat cross" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case became somewhat cross = got ones dander up.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.