The most common and most likely answer to this crossword clue is the 10 letter word BENEVOLENT
I've seen this crossword clue in The Times
B
E
N
E
V
O
L
E
N
T
More clues leading to the result BENEVOLENT
- Kindly little man takes rodent into hospital department
- Inclination to pack eastern novel, after review, friendly
- Good to live with new love at sea, getting space and time
- Well-meaning little boy taking rodent into hospital department
- Caring tendency to hide love rejected by nurse
- Been disturbed by brutal assassination of one liberal
- Friendly event noble recollected
- Kind of live appliance returned fast time
Buddy explains!

The straight part
We know that "Kindly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case kindly = benevolent.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 BENEVOLENT mean?
BENEVOLENT adjective- Having a disposition to do good.
- Possessing or manifesting love for mankind.
- Altruistic, charitable, good, just and fair.