This might help to narrow the search, although bear in mind that this is from records that are not complete and is based on assumptions. It also a long shot!

Catherine is likely to have become pregnant by someone local to her, ie in Longton.

If you search the military records on Ancestry for men in the military who died in 1918 and were either born in Longton or whose residence was Longton, then there are only 10 listed.

Local newspapers of the time often had obituaries of local men who died in the war, including photos of them. There may also be photos of the men when they enlisted. It may be possible to see a family likeness.

You may be able to find exactly where the 10 men were living, through their military records, censuses etc. The nearer to Catherine's home, the likelier they are to be the missing father. Maybe one of them was a baker or in a related industry.

Obviously, none of this would be conclusive, but it might possibly help to narrow your search.