The most common and most likely answer to this crossword clue is the 5 plus 6 letter word AGONY COLUMN
A
G
O
N
Y
C
O
L
U
M
N
Buddy explains!
The straight part
We know that "Press part that’s tender, offering pain relief?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case press part that’s tender, offering pain relief? = agony column.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.