The most common and most likely answer to this crossword clue is the 8 letter word UMBILICI
I've seen this crossword clue in The Guardian
U
M
B
I
L
I
C
I
More clues leading to the result UMBILICI
Buddy explains!

The straight part
We know that "Bits of midriff" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case bits of midriff = umbilici.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 UMBILICI mean?
UMBILICI noun- Navel
- Hilum
- A depression or opening in the center of the base of many spiral shells.
- Either of the two apertures in the calamus of a feather.
- A tube connecting an astronaut or spacecraft to the mothership, through which supplies and samples can be transferred.
- One of the foci of an ellipse or other curve.
- A point of a surface at which the curvatures of the normal sections are all equal to each other.
- An ornamented or painted ball or boss fastened at each end of the stick on which manuscripts were rolled.