The most common and most likely answer to this crossword clue is the 11 letter word TOILET-ROLL
I've seen this crossword clue in Financial Times
T
O
I
L
E
T
-
R
O
L
L
Buddy explains!
The straight part
We know that "with which to smear first half of 14" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with which to smear first half of 14 = toilet-roll.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.