It may well be worth looking at the Parish Registers or the Quarter Session records.
It is maybe a little late to find filiation orders or b@stardy orders as they were called earlier. The woman may have had to have a verbal examination before the poor law administrators to give detail on the father, who may then have been ordered to support the children, as I say, it may have been a little late in the century for that.
The parish registers may show an entry by the incumbant that names a likely father, often the case, it depends on where they lived, if in the city of london then it is unlikely, as church attendance was dropping rapidly, but if in a rural parish or small town, quite possibly there is a notation in the register.
The Poor Law Union that ran the workhouse may also have an entry in their admissions register, which should survive in the county record office covering that poor law administration district.
One other way is to see if any of the children gave a fathers name other than their mothers late husband on their marriage certificate, but again, it is an expense if you are not sure it contains any info, what you could do is ask the registrar's clerk to only send the marriage certificate if the information on it includes a father's name.