Mouthy type admits turning secret agent involved in the construction of The Wall (6)
I believe the answer is:
gypsum
'involved in the construction of the wall' is the definition.
I can't tell whether this defines the answer.
'mouthy type admits turning secret agent' is the wordplay.
'mouthy type' becomes 'gum' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'admits' indicates putting letters inside.
'turning' says the letters should be written in reverse.
'secret agent' becomes 'spy' (spy is a kind of secret agent).
'spy' reversed gives 'yps'.
'gum' placed around 'yps' is 'GYPSUM'.
Can you help me to learn more?
(Other definitions for gypsum that I've seen before include "Mineral used in making plaster of Paris" , "Kind of chalk used to make plaster of Paris" , "Plaster of Paris is made from this chalk-like substance" , "Base of plaster of Paris" , "Spy mug made from this white mineral" .)