Pet with brown tail mostly missing — odd (7)
I believe the answer is:
tantrum
'pet' is the definition.
(I've seen this before)
'brown tail mostly missing odd' is the wordplay.
'brown' becomes 'tan' (tan is a kind of brown).
'mostly missing' indicates taking the first letters (almost all of the word is removed).
'odd' becomes 'rum' (rum can mean strange or dubious).
The first letter of 'tail' is 't'.
'tan'+'t'+'rum'='TANTRUM'
'with' is the link.
(Other definitions for tantrum that I've seen before include "throwing this?" , "fit of bad temper" , "Pet, temper" , "Fit of ill-temper" , "Spoilt brat's fit" .)