Adcock was born in Auckland, but spent the years between 1939 and 1947 living and studying in England. She is a sister to Marilyn Duckworth. She studied Classics at the Victoria University of Wellington, graduating with a M.A.. She worked as an assistant lecturer librarian at the University of Otago in Dunedin until 1961. She was married to two famous New Zealand literary personalities. In 1952 she married Alistair Campbell, and later divorced. Then in 1962 she married Barry Crump, divorcing in 1963.
In 1963, Adcock returned to England and took up a post as librarian at the Foreign and Commonwealth Office in London. Apart from a brief return to New Zealand in 1975-1976, she has lived in East Finchley, north London ever since, teaching and working as a freelance writer.
Adcock's poetry is typically concerned with themes of place and everyday activities, but frequently with a dark twist given to the mundane events she writes about. Formally, her early work was influenced by her training as a classicist but her more recent work is looser in structure and more concerned with the world of the unconscious mind.