Covering the lady with the doodah (9)
I believe the answer is:
sheathing
'covering' is the definition.
(sheathing is a kind of covering)
'lady with the doodah' is the wordplay.
'lady' becomes 'she' ('she' refers to a woman).
'with' says to put letters next to each other.
'the doodah' becomes 'athing' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'she'+'athing'='SHEATHING'
'the' acts as a link.
Can you help me to learn more?
(Other definitions for sheathing that I've seen before include "putting it in a case" , "protective covering" .)