The most common and most likely answer to this crossword clue is the 4 letter word GOSH
I've seen this crossword clue in The Guardian
G
O
S
H
More clues leading to the result GOSH
Buddy explains!

The straight part
We know that "My word/a hospital that helps sick children" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case my word/a hospital that helps sick children = gosh.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.
What does GOSH mean?
GOSH interjection- (minced oath) A mild expression of surprise or enthusiasm.