Record keeping didn't arrive in western New York until the mid-19th century, a little more than 200 years after Connecticut figured it out. Up to 1850 or so, quite a few births in this neck of the woods were under the radar.
If this woman was Catharine Whaley, as she appears in the census, I've found no record of her.
Cut to January 5, 2019. I'm now convinced that the census taker wrote down the wrong surname. I believe the woman who gave birth to my great grandfather and 3 other sons by Henry Osborne was Catharine Whalen, not Whaley. There were Whaleys and Whalens in the area.
Catharine Whalen was a daughter of Jeremiah Michael Whalen and Martha Mitchell Whalen. I find them in Lima, perhaps it was South Lima, which was Henry's immediate neighborhood. They are in the 1840, 1850, 1855, and 1860 census. Jeremiah died in 1864, and the family split up. I've traced them all. Based on DNA matches to 2 of Catharine Whalen's siblings, I think I've finally found my gg-grandmother.
After the birth of her last son, Vernon Pemberton Osborne, there is no trace of Catharine. Three of her sons found work up in Rochester. She lived with none of them. Nor did she live with William, her first son, who remained in Livonia Center. I haven't found mention of her in back issues of The Livonia Gazette, whose writers kept track of everyone in town.
Vernon Pemberton Osborne bears the name of a young man who lived across the road from the Osbornes. Why would that be? At least one researcher believes that the man across the road was Vernie's father.