The most common and most likely answer to this crossword clue is the 6 plus 4 letter word DONKEY WORK
I've seen this crossword clue in The Daily Telegraph
D
O
N
K
E
Y
W
O
R
K
Buddy explains!
The straight part
We know that "Plodding that took couple to Bethlehem" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case plodding that took couple to bethlehem = donkey work.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.