The nearest Manx connection I can find is Ada Iniff's great grandmother, Ellen Rawlinson nee Surname Unknown, age 58 in the 1861 census.
Ellen's daughter, Ann, married William Bradley, a butcher b Chorley, date and place not yet established. Their 3rd child was Susannah Bradley, who was known as Anne in 1871. This is the Annie who married Thomas Iniff in 1876.
Below are the details I've uncovered. (There are several trees on Ancestry which wrongly have Thomas Iniff's wife as Annie Ackroyd Bradley.)
Ada Iniff's parents' marriage:
St Bride's LP, 14 Nov 1876
Thomas Iniff, full age bachelor, painter, 17 Chesnut St, fa Wm Iniff, painter.
Annie Bradley, 19 years with consent, spinster, 17 Chesnut St, father Wm Bradley, butcher.
Both signed their names.
Witnesses: Stephen Bradley and Eliza Boulton.
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1861
Lancashire, Liverpool, Mount Pleasant, 17h/32, household 163
20 Dale St
Ancestry's transcription:
Name Age
William Bradley 45
Ann Bradley 28
William Bradley 7
Stephen Bradley 6
Susannah Bradley 4
Robert Bradley 2
Ellen Rawlinson 58
WB 45 butcher b Chorley, Lancs
Ann+3 eldest chn b Liverpool
Robt b Toxteth Park
ER 58 mother in law, mar, b Isle of Man
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1871
Lancashire, West Derby, district 43, image 7, household 31
19 Harding St
Ancestry's transcription:
Name Age
William Bradley 50
William Bradley 17
Stephen Bradley 15
Anne Bradley 10
Robert Bradley 11
Eliza Bradley 9
Wm Bradley wid butcher
Anne's age is 13, not 10
all b Toxteth Park
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1881
Lancashire, West Derby, West Derby (Municipal), 22/16, households 76+77
4 Walpole St
William Bradley head wid 60 butcher b Chorley
John son unm 25 joiner b LP
Robert son unm 22 carter b LP
Eliza daughter unm 19 dressmaker b LP
same address:
Thomas Iniff head mar 32 painter b LP
Annie wife 24 b LP
William son 7 months b LP
Annie dau 3 b LP
Mary dau 2 b LP
(NB I think that Stephen Bradley was John Stephen Bradley, birth registered Sep Q 1855 West Derby )
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baptisms:
Robert James son of William and Ann Bradley, Toxteth, butcher. 8 Feb 1859, LP St Peter.
Eliza Ellen dau of Wm and Ann Bradley, Dale St, butcher, 4 May 1861 LP St Peter.
Jean C