Shy about Republican and awfully dour material (8)
I believe the answer is:
corduroy
'material' is the definition.
(I know that corduroy is a type of material)
'shy about republican and awfully dour' is the wordplay.
'shy' becomes 'coy' (coy can mean shy or modest).
'about' indicates putting letters inside.
'republican' becomes 'R' (abbreviation).
'and' says to put letters next to each other.
'awfully' indicates an anagram.
'dour' anagrammed gives 'duro'.
'r'+'duro'='rduro'
'coy' going around 'rduro' is 'CORDUROY'.
(Other definitions for corduroy that I've seen before include "Heavy ribbed fabric" , "Cut pile cotton fabric" , "Cotton cloth with velvety ridges" , "Material" , "Thick-ribbed trouser fabric" .)