Bit about me being reined in (5)
I believe the answer is:
tamed
'reined in' is the definition.
'tamed' can be an answer for 'reined' (I've seen this before). I am unsure of the 'in' bit.
'bit about me' is the wordplay.
'bit' becomes 'tad' (both can mean a small amount).
'about' is an insertion indicator.
'tad' going around 'me' is 'TAMED'.
'being' is the link.
Can you help me to learn more?
(Other definitions for tamed that I've seen before include "Reined" , "Domesticated or subdued" , "now behaving himself?" , "Of animals, domesticated" .)