Bower Eliza Doolittle's refuge? (6)
I believe the answer is:
arbour
'bower' is the definition.
(synonyms)
I don't understand the rest of the clue.
(Other definitions for arbour that I've seen before include "Shady garden alcove" , "Secluded garden alcove" , "'Bower, pergola (6)'" , "Sheltered leafy glade" , "Shady alcove; bower" .)