You may already be aware of this, Jasmine, but someone has already input a good bit of information about what appears to be your family at the FamilySearch site: https://www.familysearch.org It's run by the Mormon church and has the largest database of genealogical info in the world. You have to register to use it, but it's entirely free. If you look there, you'll find Anna Dougher at her unique ID number, which is LR3S-2CL. They have her mother listed as Anna as well, which is sometimes found as an alternate to Honora or Nora in various records. The records show someone listed as "Dooherman" who input much of the info, so he might be a relative with whom you could compare notes (you can private message him at the site). Unfortunately, there's no further info about their Irish origins, apart from Mayo being mentioned in some records, but the Pennsylvania records appear to be extensive..
My motther's parents grew up in a part of Mayo a bit more to the east of where your Dougher's may have come from (mentioned above), so I'm somewhat familiar with the parish registers in the area. They are available online and you can download each page and study them at your leisure. As an example, heres a link to the register for the parish of Kilmoremoy, where the records go back to 1823: https://registers.nli.ie/parishes/0753
Once there, you can use an interactive map to check neighboring parishes. Since you have likely years of birth, it should not be too hard to locate baptismal records, at least if they occur in parishes with old enough records. Some other nearby parishes with records going back far enough for what you want are Crossmolina (1831), Kilfian (1826), and Addergoole (1840).