Gail, you probably already have the following info, but for the benefit of Michael and others, to save covering the same ground for background info to aid the search :
Marriage at Eglingham :
1810. June 14. Richard Mitchinson and Alice Clark. Witnesses James Clark, Thomas Richardson, Ann Scott.
Baptisms at Wooler Presbyterian :
1811. Born 24 Jan, baptised 13 Feb. John Mickison, son of Richard and Ellison.
1813. Born 5 Jan, baptised 26 Jan. Jane Mitcheson, daughter of Richard and Elison.
Baptism at Clayport Presbyterian Alnwick.
1815. Archbold, 2nd son of Richard and Allison Mitchison. Born 5th, baptised 23rd January.
Burial at Embleton Holy Trinity.
1817. 23 Feb, Richard Mitchison, 28, of Stamford.
Baptism at Burgher Secession Chapel, North Sunderland.
1817. Margaret Richard, daughter of Richard Mitcheson, Husbandman in Stamford, and Alice his wife, was born 24th, baptised 27th Feb.
So Margaret was born the day after Richard was buried, hence her middle name.
Oldest son John and youngest daughter Alice (can't find her baptism) are together in Lucker in 1841 as stated, no Alice Snr.
Oldest daughter Jane is with husband Alexander Davidson in Ancroft in 1841, no Alice Snr. HO107/318/1/34/19
2nd son Archbold is with his wife Ann (Bolton) in Ellingham in 1841, no Alice Snr.
HO107/819/1/ /1
2nd daughter Margaret is with her husband William Patrick in Newcastle in 1841, no Alice Snr. HO107/848/2/7/7
So she's not with her children, unless there are any others.
Alan