Hi
I cannot find the birth on the internet. You said that the only birth that could match is in Basford. You could be right but there may be the correct birth that you haven't found yet.
My advice is to go to the Derbyshire Look Up board and ask for a Birth look up for Francis/Frances Clayton, 1845-1847, Derbyshire. You could narrow it down to Shardlow or Basford or leave it open for all of Derbyshire.
The people doing the look ups have far more resources than the ones available on the internet, eg Family History Society CDs, IGI. You may be lucky and find it.
Good luck
Jane