If Jessie was born around 1874 she should appear in Civil records, however I'm not seeing any good match for her on the BMD Index on FamilySearch. She could have been registered before a name was decided, so could appear as 'Female' Cross.
Do you have her mother's maiden name - maybe from her marriage or death cert ?
Here's a list from the BMD Index for Cross births around 1874. Cork is a large county, so there are a number of districts that cover the county - i.e. : Cork (based in the city), Bandon, Bantry, Castletown, Clonakilty, Dunmanway, Fermoy, Kanturk *, Killmallock *, Kinsale, Lismore, Macroom, Mallow, Middleton, Millstreet, Mitchelstown, Skibbereen, Skull, and Youghal *https://www.familysearch.org/search/records/index#count=20&a...
(the districts with * cover parts of other counties as well as Cork)