The most common and most likely answer to this crossword clue is the 4 plus 4 plus 5 letter word KEEP OPEN HOUSE
I've seen this crossword clue in The Daily Telegraph
K
E
E
P
O
P
E
N
H
O
U
S
E
Buddy explains!
The straight part
We know that "Offer hospitality to all" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case offer hospitality to all = keep open house.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.