What makes you suspect her name could be Hawker?
Her childrens births show her maiden name as Harker, she married as Harker (dau. of George Harker) and although she states bc.1847 Houghton le Spring on 1871 census she was in fact born in 1849 - Alice Harker birth Dec.1849 - mmn Cummings.
In 1851 she is with her parents George Harker and wife Ann in East Rainton, Durham.
Her surname was definitely Harker.
Annette