If you can give us a better idea of what you already know and what you would like to find it will be easier for us to help you.
Do you have Hamilton Wray's marriage(s)? father's name and occupation should be listed.
Also, keep in mind variations of the surnames Wray (Ray, Rea, etc.) and Lemon (Lamont, Lamond, etc.).
A Hamilton Wray married 1859 (Dungannon district) Elizabeth
A Hamilton Wray married 1877 (Dungannon district) Margaret
A Hamilton Wray married 1878 (Lurgan district) Margaret
Moyroe townland is in Dungannon registration district and if the Hamilton Wray in Moyroe in 1911 census is the correct person then this looks like his son William John's birth in 1892- mother
Margaret Malliaghhttps://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1892/02362/1882247.pdfIn the 1901 census Hamilton Wray's wife is Margaret so it doesn't look like he married Mary Jane Lemon unless the marriage was at a much later date?