Right-winger in Barking breaking valuable material is monstrous type (4)
I believe the answer is:
ogre
'valuable material is monstrous type' is the definition.
I can't tell whether this definition defines the answer.
'right-winger in barking breaking' is the wordplay.
'right winger in' becomes 'og' (I am not sure about this - if you are sure you should believe this answer much more).
'barking' means to remove the middle letters (I've seen 'bark of' mean this (the outside letters or 'bark' of the word)).
'breaking' becomes 'rupture' (rupture is a kind of breaking).
'rupture' with its middle taken out is 're'.
'og'+'re'='OGRE'
Can you help me to learn more?
(Other definitions for ogre that I've seen before include "character in 'Puss in Boots'" , "Brute" , "Fairy tale giant" , "Wicked and inhuman person" , "Giant of folklore who ate humans" .)