Many thanks to you all for your welcome information. On the GRO online index there are no recorded births in the name of Green with MMN of Carter in Sleaford or the surrounding area in the 1860's/1870's. Ann's given surname of Carter on the 1863 marriage certificate can be linked to the MMN of 'Cart' on two of the children's births - William Edward & Ann Keziah.
All of your research does strongly suggest that the 1863 marriage was that of Edward Green and Ann Curtis, not Carter. A further thought is that the marriage banns might have her surname recorded correctly.
Rosie, could you kindly advise the name of the church where they were married, and the date, from the marriage certificate.
Geoff