Ken, I looked at your tree on Ancestry for clues, and note that you have Ann Jane Lamothe's mother's name as Jane Archibald, b circa 1810, Scotland. You have Jane's first marriage to a John Keigg producing Keigg children in the IOM, then as a widow, a relationship with a Mr Lamothe producing Ann Jane b 1857 in England, and finally, a marriage to a Mr Radcliffe.

Starting with the two censuses in which Ann Jane's mother is easy to identify (using information from your tree):

*****
1891
Andreas 3b/2
St Jude's Rd
Ancestry has mistranscribed Galloway as Eahaway:
Name Age
W Eahaway 37
Ann Jane Eahaway 34
Harriet J Eahaway 10
Janet G Eahaway 9
William Eahaway 7
James W Eahaway 5
Elizabeth Eahaway 3
Charles Eahaway 0
Jane Radcliffe 74

JR 74 mother in law wid living on own means b ANDREAS
*****
1901
Scotland, Eastwood Burgh, Eastwood, Renfrewshire
39 Stevenson Drive
Name Age
William Galloway 46
Ann J Galloway 44
Janet Galloway 19
William Galloway 17
Wilfred Galloway 15
Elizabeth Galloway 13
Charles Galloway 11
Lina Galloway 8
Dora Galloway 4
Mildred Galloway 2
Baby Galloway 1wk
Jane Raddiffe 84

JR 84 grandmother, living on own means, b ISLE OF MAN
*****

---- Ann Jane's mother seems likely to have been a Manxwoman b circa 1817.

I haven't found Ann Jane in 1861, but she appears as Ann J Archibald in 1871, living with her half brother, James Edward Kegg :

*****
1871
Maughold 1b/6
46 Parliament St
Ancestry's transcription:
Name Age
James E Kegg 29
Catherine Kegg 36
Thomas E Callow 13
Arthur B Callow 11
Albert H Callow 3
Eliza J Callow 8
Ann J Archibald 14

JEK 29 Hostler b Lez
CK wife b Andreas
Callow step chn b Lez+And
AJA sis b England

******
James's marriage from the new IGI:
Name: James Edward Kegg
Event: Marriage
Event Date: 12 Apr 1870
Event Place: Lezayre, Isle of Man
Gender: Male
Death Age:
Marital Status: Single
Father: William Kegg
Mother:
Spouse: Catherine Callow
Spouse's Marital Status: Widowed
Spouse's Father: Thomas Quayle
Spouse's Mother:
Digital Folder Number: 004493956
Image Number: 00172

********
and his baptism:
Name: Jas Edwd Keig Archibald Keig
Event: Christening
Event Date: 20 Apr 1841
Event Place: Lezayre, Isle of Man
Gender: Male
Death Age:
Marital Status:
Father: Wm Keig
Mother: Jane Archibald Keig
Spouse:
Spouse's Marital Status:
Spouse's Father:
Spouse's Mother:
Digital Folder Number: 004493705
Image Number: 00096

******
Jane's first marriage:

Name: William Kegg
Event: Marriage
Event Date: 09 Jul 1837
Event Place: Lezayre, Isle of Man
Gender: Male
Death Age:
Marital Status: Single
Father:
Mother:
Spouse: Jane Archibald
Spouse's Marital Status: Single
Spouse's Father:
Spouse's Mother:
Digital Folder Number: 004493823
Image Number: 00195

*****
Jane's second marriage:
Name: Thomas Radcliffe
Event: Marriage
Event Date: 05 Oct 1861
Event Place: Lezayre, Isle of Man
Gender: Male
Death Age:
Marital Status: Widowed
Father: Thomas Radcliffe
Mother:
Spouse: Jane Kegg
Spouse's Marital Status: Widowed
Spouse's Father: Charles Archabold
Spouse's Mother:
Digital Folder Number: 004493956
Image Number: 00102

******
I can't find Jane's Kegg family in 1841, but here she is in the following censuses:
*****
1851
Lezayre 2/13
Dry Close (next to Close Churm)
Keig mistranscribed as Reig
Name Age
Jane Reig 33
John Reig 13
William Reig 11
James Edwd Reig 9
Thomas Reig 7
Daniel Reig 5
Joseph Reig 3
Elizabeth Quayle 53

JK 33 wid farms 13 acres b Lez
chn all b Lez
EQ servant

****
1861
Lezayre 4/11
4 Close Churm
Name Age
Jane Keigg 43
Edwd Keigg 19
Daniel Keigg 15
Joseph Keigg 12
Edith Quayle 64

JK 43 wid farming 10 acres b Lezayre
Elizth Quayle (mistranscribed as Edith) relative, unmarried, lodger b Lezayre
chn all b Lez

*****
1871
Lezayre, district 4+6, image 3
Dry Closes
Thomas Radcliffe head mar 73 farmer b Andreas
Jane wife 52 b Lezayre
Elizabeth Quayle wife's mother UNMARRIED 74 boarder b Lezayre
*****
************
1881
Lezayr D 4c/12
Dry Closes
Thomas Radcliffe head mar 85 farmer of 9 acres b Andreas
Jane wife 65 b Lezayre
Catherine grdau unm 23 domestic servant b Andreas
John R Kneale grson 2 b Lezayre

*****
Jane's illegitimate birth ?

Name: Jane Quayle
Event: Christening
Event Date: 11 May 1817
Event Place: Lezayre, Isle of Man
Gender: Female
Death Age:
Marital Status:
Father:
Mother: Elizabeth Quayle
Spouse:
Spouse's Marital Status:
Spouse's Father:
Spouse's Mother:
Digital Folder Number: 004493705
Image Number: 00037

*****
Great news that Mary M has confirmed the name of Ann Jane's father !

Jean C