The following is a bit of a long shot but have you looked at the entry on the 1881 census for the household of James Quirk aged 67 from Rushen. The IGI lists his household as containing his granddaughter Annie J Quirk aged 4, and also his daughter Ann Yate, a mariner's wife aged 21 and her son William Yate aged 4 mths. For this to be the correct entry it depends on handwriting being misread, an I for a J and a T for a Y, and also for Annie to have lied about her age in later life. Not uncommon especially if wanting to marry underage. Normally, I would be hesitant to suggest such a complete stab in the dark, but looking on the Lawson's birth records I have found a William Tate born to Edward Tate and Ann (nee Quirk) in November 1880. The census was taken April 1881. I cannot find the name Yate anywhere. Sorry if this is completely off, but if stuck may be worth a look. She has to be somewhere. Good luck.