The most common and most likely answer to this crossword clue is the 3 plus 4 plus 4 letter word AND THEN SOME
I've seen this crossword clue in The Sunday Telegraph
A
N
D
T
H
E
N
S
O
M
E
More clues leading to the result AND THEN SOME
Buddy explains!
The straight part
We know that "to put it mildly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case to put it mildly = and then some.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.