I have now checked the marriage of Joseph & Elizabeth on the St Johns Coll transcripts but it doesn't give any further info, however he was given of Wooten, there is only one baptism for a Joseph Badcock at Wooten in the right sort of time frame (though there are others in nearby areas)
1704 Joseph s/o Joseph & Jane Badcock
- they also had a daughter Jane in 1702, but I can't see any other children for them.
Joseph Badcock of Wooten married Jane Harding of Wooten 3rd Oct 1700 in South Hinksey
Burials from Wooten
1750 Mary d/o Joseph & Elizabeth Badcock
1753 Jane wife of Joseph Badcock
1762 Joseph Badcock
.
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1773 Joseph Badcock of Marcham
1776 Elizabeth Badcock widow of Joseph of Marcham
Looks possible/likely

DebbieG