The most common and most likely answer to this crossword clue is the 5 plus 3 plus 5 letter word PRESS THE FLESH
I've seen this crossword clue in Financial Times
P
R
E
S
S
T
H
E
F
L
E
S
H
More clues leading to the result PRESS THE FLESH
Buddy explains!
The straight part
We know that "to solicit support" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case to solicit support = press the flesh.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.