The most common and most likely answer to this crossword clue is the 3 plus 7 letter word DAY NURSERY
I've seen this crossword clue in The Daily Telegraph
D
A
Y
N
U
R
S
E
R
Y
More clues leading to the result DAY NURSERY
Buddy explains!
The straight part
We know that "Care centre with no overnight charges" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case care centre with no overnight charges = day nursery.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.