Divorce in the period you are interested in was rare because it was so expensive. There was no legal aid available until after the First World War so it was prohibitively expensive for most working people. Though men could get divorced on the grounds of adultery alone, women needed more, so an abusive husband would not be enough reason for a woman to divorce a husband and anyway proving abuse would also be difficult. A woman would need to prove adultery and something else - desertion being the easiest.
Most people when such a marriage broke down had to be practical if they wanted a new relationship. As divorce was beyond their reach they lived with their new partner as man or wife or in some cases they bigamously married (suprisingly common particularly where all parties were in agreement after the breakdown of the marriage or if one moved away and lived anonymously in a large city).
So the first thing to consider might be that a marriage didn't take place between Elizabeth Hancox and Frederick Jefferies, or it did quietly many years later when Elizabeth was able to marry because of the death of her first husband, or because the divorce laws had eased after the First World War (though they might not chose to divorce then because that would be to admit publically they were not married in the first place and their children were illegitimate - children born before a marriage could only be legitimised by a subsequent marriage if they were born after the law which allowed this in 1926).
I have one such case where the couple were living together as a married couple in the 1890s did not lmarry until 1924.
The fact that Elizabeth Hancox' s children's birth certificates say her name was Hancox - that should be her maiden name. If she had another name since her maiden name then you would expect it to read
Elizabeth Jefferies late ----------- formerly Hancox. The formerly name being her maiden name.
If you are still convinced there was a divorce The National Archives holds all the divorce records for this period and you will find their research guide here
http://www.nationalarchives.gov.uk/catalogue/researchguidesindex.aspscroll down to Divorce Records after 1858.
Have you found the couple on the 1901 census? Do you know anything else about Elizabeth that might allow a search for her on the 1891 census?
Regards
Valda
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