Hi Retsina,
Mary White (mother of Ann King b1816) was indeed the daughter of Thomas White, Farmer, & Jean Smith [per Mary White's 1865 Death Cert, which also states that Mary died at Giffen, and that she was widow of James King, Farmer].
I've been researching your Ann King's oldest brother, James King b1802, who farmed at Lugtonridge, Beith. You'll find him there in the 1841 & 1851 Census. James married Margaret McIntyre in 1849 and subsequently spent time in Ireland, and Kilmory, Bute where his wife originally came from.
James' 1802 birth record describes him as son of Robert King, Lugton, and Mary White, so it would seem the King family had been farming at Lugtonridge for a long time. Beware though that there were 6 farms all adjacent to each other, 5 of which were all named Lugtonridge, with the 6th farm named High Lugton. The Lugtonridge farms are located in the east of Beith right next to the border with Dunlop parish.
The 1801 Beith Parish Marriage record for Robert King & Mary White describes them as "both in this parish", yet the births/baptisms of all their 10 children (including my James and your Ann) are recorded in Dunlop parish - perhaps because the town and church in Dunlop were physically much closer to Lugtonridge than the Beith church.
The reason I mentioned multiple farms named Lugtonridge is because another of those farms was farmed by a Robert King (c1770-c1852), who married Janet McCulloch in Neilston, Renfrew in 1798 and had multiple children between 1799-1814 (this family also appears in 1841 & 1851 Census at Lugtonridge). This might be a more distant relation but it certainly isn't our Robert King who was having children with Mary White at the same time. In fact because son James King was already Head of the household in 1841 Census, with his mother Mary residing with him, I suspect our Robert had already died by then - although I haven't found his death. [FamilySearch Wiki for Beith indicates that parish Death records only cover the period 1783-1787 so I don't expect to find his death unless a Memorial Inscription turns up].
I hope this is of some help. Don't hesitate to PM me if you want to exchange further info.