The most common and most likely answer to this crossword clue is the 6 letter word CARESS
I've seen this crossword clue in The Times
C
A
R
E
S
S
More clues leading to the result CARESS
- Burdens extended at the end? Show sign of affection
- Hug little son weighed down by worries
- Pet frightens with head right down
- Is concerned about son's embrace
- Stroke is worried about onset of storm
- Starts with leader moving to the back— that's a nice touch
- Worries over small pet
- Having tail extended worries pet
- Worries over small stroke
- Fears first to last to show affectionate touch
Buddy explains!

The straight part
We know that "Pet" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case pet = caress.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 CARESS mean?
CARESS noun- An act of endearment; any act or expression of affection; an embracing, or touching, with tenderness.
- A gentle stroking or rubbing.
- To touch or kiss lovingly; to fondle.
- To affect as if with a caress.