In 1901 William BARTLAM and Sarah appear to have already separated, and as Violet born abt 1900 has been adopted out it looks like the separation may have been round about Violet's birth in 1900. Sarah's next child is Mary Eliza Yarwood BARTLAM in 1904. 4 years is quite a gap, so it reduces the likelihood or John YARWOOD being the father.
If all this does turn out to be the correct family, then maybe DNA might prove things. Another thing to try is photos. I've just seen one of my distant relatives on Facebook for the first time. He is my 2nd cousin twice removed. Both my husband and I said straight away "He looks like Auntie Elsie!". He may not be flattered, but the resemblance is striking!