It's funny: just as Sophia runs away from home, finds Tom at the inn at Upton, and then decides to travel to London to stay with her relative, Lady Bellaston, Harriet Fitzpatrick suddenly turns up. Harriet is Sophia's cousin, of roughly the same age. They played together as children, calling each other "Miss Graveairs" (since Sophia has always been serious and moral) and "Miss Giddy" (since Mrs. Fitzpatrick has always loved fun). And now, here she is, riding along the same path away from the inn at Upton.
Why does Mrs. Fitzpatrick turn up right in the novel while Sophia is leaving Tom behind? Because Mrs. Fitzpatrick is Sophia's perfect foil. She's like a physical representation of the road not taken. When Sophia first leaves her father's house behind, she does it because she does not want to be forced into marriage against her will with Mr. Blifil.
But she manages to find Tom as she's traveling, and she almost reunites with him without her father's consent. That would have been a problem, since Squire Western would probably have disinherited her forever, and Squire Allworthy would certainly never have forgiven Tom. Tom's own hyper-sexed ways prevent that from happening, which turns out to be a good thing.
But Mrs. Fitzpatrick is an example of what might have happened if Sophia and Tom had eloped. Like her cousin, Mrs. Fitzpatrick spends several years hanging out in Bath with her aunt, Mrs. Western, when she reaches young adulthood. But unlike Sophia, Mrs. Fitzpatrick spends all of her time out at parties hanging out with boys. One of these boys is an Irishman named Mr. Fitzpatrick. Mrs. Fitzpatrick soon falls head-over-heels for him. She runs away with him, and the two get married. But it's only once this binding contract passes between them that Mrs. Fitzpatrick sees Mr. Fitzpatrick's true colors.
All of the consequences of Mrs. Fitzpatrick's foolish marriage—her alienation from her family (since Squire Western and Mrs. Western both refuse to talk to her ever again, under any circumstances), her isolation in Mr. Fitzpatrick's home, and her relative lack of options when she realizes that he is a cheating, thieving jerk—leave Mrs. Fitzpatrick in a terrible position.
What's more, the novel appears to judge Mrs. Fitzpatrick's desertion of her husband to be a bit immoral. (This judgment seems to us to show the times in which the novel was written; to us, the fact that Mr. Fitzpatrick locked Mrs. Fitzpatrick in a room for weeks to try and forceher to sign over some of her property to him is excellent grounds enough for Mrs. Fitzpatrick to run away.)
Sophia even advises that Mrs. Fitzpatrick find some way to reconcile with her husband, since a wife at war with her husband "can hardly make a disadvantageous peace for herself on any conditions" (11.10.11). And of course, Sophia also notes that Mrs. Fitzpatrick is a very good liar—another sign that the book is judging her as having very little moral fiber. Mrs. Fitzpatrick attaches herself to a nameless "Irish nobleman," in much the same way that Mrs. Waters lives with Captain Waters. It's this Irish nobleman who supports Mrs. Fitzpatrick when she decides to ditch her husband, an arrangement that Sophia definitely does not approve of.
What Mrs. Fitzpatrick learns once she arrives in London is that there is no turning back, now that she has married Mr. Fitzpatrick without her family's consent. She tries to repair her relationship with Squire and Mrs. Western by writing to them with Sophia's location in London. But even though she helps them, Mrs. Western still gives Mr. Fitzpatrick information that leads him to his runaway wife, all while swearing that she will never, no matter what, speak to Mrs. Fitzpatrick again.
Now, we don't think the novel is suggesting that it is a good thing that Mrs. Western has cut off her niece without a word. But the novel does seem to find her behavior understandable, at least. At any rate, Mrs. Fitzpatrick certainly never patches things up with the Westerns. She tries to get her revenge on unforgiving Mrs. Western by encouraging Tom to make Mrs. Western fall in love with him, but when Tom refuses that plan (and Mrs. Fitzpatrick's own sudden attraction to him), she more or less drops out of the novel.
By the end, we find out the outcome of Mrs. Fitzpatrick's disastrous marriage: the two Fitzpatricks wind up separated. Mrs. Fitzpatrick finds a place to live in London, where she lives off what remains of her fortune. The novel cattily notes that Mrs. Fitzpatrick "maintains a perfect intimacy with the lady of the Irish peer; and in acts of friendship to her repays all the obligations she owes to her husband" (18.13.15). This is yet another jab from Fielding at the immorality of London high society, since none of the people involved think it's a problem that Mrs. Fitzpatrick is good friends with the wife of the man with whom she is having an affair.
This final note about Mrs. Fitzpatrick proves that her experiences with the world, and her decisions to run off with men periodically, have left her permanently morally damaged. Sophia only avoids that fate by staying true to Tom but waiting until they have their parents consent to marry him.