Presumably the mother is known, as is the childs name and the dates/area etc. With that info, one could do a bit of inferring and logical deduction, to reduce the pool of information to more likely people.
Please see reply 8 - she was living with her parents in 1861 and was married by 1871. The child is shown as "wife's" on the 1871 so clearly not her husbands child
We cant be sure about that.
"The husband could be the childs father but on the 1871 census the child is described as 'child of wife'" from reply 8.
If she is re-married by 1871 her previous surname may be on the marriage cert. It may be another route to search.
All roads lead to Rome, or a ditch with mud and water in my case.
Cheers.