Agatha Smith was the eldest daughter of Barthwell Smith with his first wife Eleanor and her baptism is as follows:

I have never been able to trace Barthwell’s daughters from his first marriage until another descendant of Barthwell Smith’s, Jo Sexton found the following marriage for a HARGETTER HEAPES, which she kindly shared with me, this clearly shows Hargetter was a widow and the daughter of Bathwell Smith, could Hargetter be another name for Agatha.

Taking the fact that Hargetter is shown as a widow in the above would lead to looking for a marriage between a Heapes and a Smith. There is a strong possibility of such a marriage in 1834 at Tissington, Derbyshire:

Is IGETTA SMITH named above simply a miss-spelling of Agatha, would also need to know what happened to Samuel Heaps.
Going back to Hargetter Heapes, The 1851 & 1871 census shows as follows, yet to find the 1861 Census return for Hargetter and John Wright.

The 1851 Census would give a birth for Hargatteer 1809 and the 1871 Census 1818 both give a birthplace as Medburn, Leicestershire. Agatha was baptised in 1816 in Medbourn cum Holt, Leicestershire.
Further records would indicate that John and Hargetter Wright had at least one daughter that I have found so far:
Name: Elizabeth Wright Gender: Female Christening Date: 14 Oct 1860 Christening Place: Leek, Staffordshire, England Father: John Wright Mother: Hargetter Wright
Hargetter Wright dies 1st Jan 1880 at 17 Trinity Street, Sheffield.

Burial Records:
WRIGHT, Hargetter (Wife of John Wright, age 62).
Died at 17 Trinity Street; Buried on January 4, 1880 in Consecrated ground;
Grave Number 100, Section Q2 of Burngreave Cemetery, Sheffield.
Using the 1849 marriage certificate showing Hargetter’s father’s name Bathwell Smith as well as the information shown leads me to the conclusion that Agatha, Igetta and Hargetter must be one and the same person. Again thanks and appreciation have to go to Jo Sexton for finding the 1849 certificate and pointing me in the direction of the above research.