Are you saying that you want to trace Edward Bonner and that all you know about him is that he is on the electoral roll from 1919-1923, and that his surname is a middle name for your father and some of his siblings?
As Annette says that 1920 marriage certificate seems a must, although whether you can believe his father's name is a moot point.
Do you know (roughly) when and where he died? Or how old he was in 1920 (again, the marriage certificate would say)?
Did she carry on at that address after 1923 and he disappear (even though a daughter was born in 1924)?
The odd thing is that, although she was free to marry him (her first husband having died) and they did marry, the children were registered under her surname. It is possible that he was already married - she may not have known that at first - and it is possible that they fell out after 1923, but that doesn't explain the birth registers.
(There is another Bonner having children in Petersfield at the same time (mmn Venes) - although I can't find their marriage)
Richard