The most common and most likely answer to this crossword clue is the 6 plus 5 letter word STRING ALONG
I've seen this crossword clue in Financial Times
S
T
R
I
N
G
A
L
O
N
G
Buddy explains!
The straight part
We know that "Offer false hope to" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case offer false hope to = string along.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.