Let’s get the book’s official statement on the subject out of the way first. Late in the novel, Harriet’s father is stated to be a tradesman, marital status unknown, who is “rich enough to afford her the comfortable maintenance which had ever been hers, and decent enough to have always wished for concealment.” Her father approves of the match with Robert Martin, and there’s a suggestion he possibly settles money on Harriet on her marriage. Nothing is said of the mother. I personally do not think Harriet is related to anyone we meet in the book. No named character in the book is high enough in status to weather the scandal of being known to have fathered or given birth to an illegitimate child. This doesn’t mean that they wouldn’t have done it, just that they wouldn’t have kept the child, and the accompanying risk of gossip and scandal, in the neighborhood of Highbury. However, the alternative theories are potentially of interest to people writing Jane Austen spinoffs, so let’s go over them. You’ll notice I don’t really address the question of whether particular character seem moral/immoral enough for certain behaviors; a lifetime of reading murder mysteries makes me unwilling to go that route in discussing what fictional people are capable of.
Some have speculated that Harriet is the daughter of Miss Hetty Bates, on the basis of the similar names: “Hetty” is usually short for “Henrietta” but could theoretically be short for “Harriet,” and both female names are variations on the male name “Henry.” Miss Bates is additionally a person of a plausible age to have a daughter, and babbles nervously in a way that looks guilty to some readers. But it’s a gossipy neighborhood. People would know if Hetty was having an affair, and they would notice if she left town for a few months because she was pregnant. They’d have been a lot more judgmental of a parson’s daughter having a child out of wedlock than the people of Highbury seem to be towards Miss Bates. And then there’s the question of whether Miss Bates could keep her mouth shut about a guilty secret for that long. I vote no.
You will see Mr. Woodhouse floated occasionally as Harriet’s “real” father, with the tradesman being a convenient fiction to keep from shocking Emma Woodhouse. You will usually see this idea in conjunction with the concept of Miss Bates as the mother and the “Dirty Old Man Woodhouse” theory, which, as previously stated, rests on a false idea of what was and was not considered socially appropriate in Austen’s world. I see only two points in favor of the “Harriet Woodhouse” idea. One, Mr. Woodhouse is very fond of the Bates women and they of him. Two, he’s the only moderately prominent male character of the right age who is dumb enough or insular enough to think keeping his lovechild in the neighborhood is a good idea. For me, the idea ultimately breaks down due to his lack of brain-mouth filter. He doesn’t babble as continuously as Miss Bates but he doesn’t seem like he’d be good at keeping a secret for seventeen years either.
I rule out the Knightley men (including their late father) on the grounds that they are not dumb enough to keep a lovechild in the neighborhood, and that George, at least, would be candid with Emma about it around the time he proposed to her. Also, it makes Harriet’s crush on Knightley really disturbing. Am not sure that John, the younger brother, is old enough to have a seventeen-year-old child out of wedlock. I rule out Mr. Weston on the grounds of being, like the Knightleys, not dumb enough to court scandal by keeping a lovechild in the neighborhood, and on the grounds of being, like Mr. Woodhouse, not secretive enough to conceal something like this. George Knightley and Mr. Weston are particularly bad candidates for Secret Lover of Hetty Bates, because they are both unmarried men of the same class as Hetty. If she became pregnant by either of them, the social expectation would have been for the man to marry her, and they seem like the kind who would comply with social expectations in that regard.
This basically leaves a couple of scenarios where Harriet is the illegitimate offspring of one of the large group of deceased parents mentioned in the story. These are all cases where we simply don’t know enough about the characters to know for sure whether they would do such a thing. Let’s run through the possible maternal candidates:
-The Late Mrs. Captain Weston, with the unnamed tradesman: timeline doesn’t work; she died in 1791 in Moody’s chronology, and the same chronology has Harriet born in 1796.
-The Late Mrs. Woodhouse, with the unnamed tradesman: Seems to have died around 1804-1805, so after Harriet’s birth. I can’t rule out her having the brains and the money to successfully conceal all this from her family and the neighborhood, especially if the original affair occurred when she was visiting friends or relatives away from Highbury, and she found an excuse to also spend the later part of her pregnancy away from Highbury. In a different kind of author’s work, Mr. Woodhouse might have been tailor-made for cuckolding. But again, I don’t see any reason why she’d want the child raised in the vicinity of her legitimate children.
-The Late Mrs. Lieutenant Fairfax, with the unnamed tradesman: it’s not clear whether she lived in Highbury while her husband was away with the military, and it’s not clear how long she lived after her husband’s death in action, although it seems to have been less than a year. A grieving widow having a tragic affair and then dying in childbirth, with the child being raising at the father’s expense, in the vicinity of the mother’s family, is painfully melodramatic even for the rather melodramatic Fairfax sidestory. And in this scenario, there’s no reason to raise the child near Highbury unless Mrs. Fairfax’s family, the Bateses, see some value in it, and then we’re back to the improbability of Miss Bates keeping a secret for seventeen-ish years
You can vaguely construct a scenario where the Late Lt. Fairfax had some kind of fling with the unnamed tradesman’s daughter. After the mother died in childbirth and the father died in combat, the maternal grandfather settled the girl near Highbury while somehow having no clue that the girl’s father’s wife’s relatives lived there. In this scenario, the maternal grandfather is allowing himself to be passed off as Harriet’s father at the end of the book, maybe to avoid awkward questions now that he knows Lt. Fairfax’s in-laws and legitimate daughter are in the area. But although there is some dramatic irony in Harriet being Emma’s half-sister (from whichever parent), there’s very little in her being Jane’s half-sister. So, this scenario is workable for fanfic purposes but should not be taken as reflective of Austen’s intentions.
If you combine Ellen Moody’s calendars for Sense and Sensibility (1797-1798) with Jo Modert’s calendar for Emma (1814-1815), it looks like it is possible for Harriet to be the offspring of Eliza Williams and John Willoughby from S&S. I personally think Brandon would look after his ward’s daughter better than that, but you never know. Again, useful for fanfic purposes, less so for analysis of Austen’s work, but that’s been the point of this post all along.

It’s going to be amazing to finally dive into all the novels with this huge level of ancillary explanations floating in my head!
Anyway, just based on your analysis and what I know about writers, it’s perfectly plausible that Jane might have kept a real, neighborhood rumor she’d heard in mind when she contemplated Harriet Smith’s background.
But, since knowing more about Harriet Smith’s background was immaterial for her novel, she left Harriet’s background open for fanfiction writers of today.
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Thank you!
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