Sources indicate the Joseph Smith sired a child through Fanny Alger:
"Chauncy Webb suggested that Emma learned about Joseph’s marriage to Fanny Alger when the girl became pregnant. According to Wilhelm Wyl, who interviewed 'Mr. W,': 'In Kirtland, [Joseph] was sealed there secretly to Fanny Alger. Emma was furious and drove the girl, who was unable to conceal the consequences of her celestial relation with the prophet, out of her house.'
"There is no record that Fanny, in fact, had a child, but Emma’s angry reaction would be consistent with her later behavior under similar circumstances. She obviously did not consider it a genuine marriage. . . .
"There are certainly a number of scenarios (including miscarriage and stillbirth) by which Fanny could have been pregnant but had no child who made it into contemporary records. In 1878, William McLellin told Joseph F. Smith and Orson Pratt: 'Emma Smith told him that Joseph was both a polygamist and an adulterer.' (Joseph Fielding Smith, 'Life of Joseph F. Smith, Sixth President of the Church of Jesus Christ of Latter-day Saints' [Salt Lake City, Utah:p. Deseret, 1938. 239]. If Emma made such a statement and if McLellin reported it correctly (he would have been 72 in 1878), then it may mean that Emma accepted Nauvoo plural marriage as 'polygamy,' but rejected Joseph’s Kirtland relationship with Alger, calling it 'adultery.'"
("Joseph Smith's Polygamy: The Joseph Smith-Fanny Alger Relationship--A Brief History," at: http://www.josephsmithspolygamy.com/FannyAlger/MASTERFannyAlger.html
Mormon historian Todd Compton, in his book, "In Sacred Loneliness: The Plural Wives of Joseph Smith" (Salt Lake City, Utah: Signature Books, 1997, p. 35), concludes that Smith likely had sex with Alger, got her pregnant and that:
--the baby was born but died prematurely, or
--the baby was raised under a different identity:
"[Chauncy] Webb acknowledges that this was a fully sexual union. Since there is no record of Fanny having a child, either Webb was mistaken (though that seems unlikely, if Fanny lived in his home after leaving the Smith home), the child was miscarried or died young, or it was raised under another name. Without futher documentation, there is no way of knowing."
Don Bradley, an LDS history specialist, presented a paper at the August 2010 Sunstone Symposium entitled, "Dating Fanny Alger: The Nature, Timing and Consequences of an Early Polygamous Relationship."
Bagley summed up his paper's thesis as follows:
"Fanny Alger left Joseph and Emma Smith�s home pregnant, under Emma�s wrath, and in the middle of the night. The incident set Kirtland on fire with rumors of the prophet�s adultery�or was it polygamy? Some scholars have argued that the relationship was an 1835 affair, too early for polygamy, others that it was an 1833 marriage. Which of these theories is right? Or are they all they wrong together? I will piece together what happened the night Fanny was evicted, what consequences followed, and when all this occurred, illuminating Joseph and Fanny's relationship and other longstanding enigmas."
(Don Bradley, abstract of paper, "Dating Fanny Alger: The Nature, Timing, and Consequences of an Early Polygamous Relationship," delivered at "Sunstone 2010 Symposium and Workshops, Salt Lake City, Utah, 7 August 2010, at: https://www.sunstonemagazine.com/wp-content/uploads/2009/05/SLC10-final-7-29-small-for-web1.pdf
Bagley's presentation was later summarized thusly:
"Bradley tried to pin down when the 'affair' happened. Apparently, Emma discovered Joseph and Fanny late at night in the barn. According to Bradley, Alger appeared pregnant. Emma threw a fit, and threw Alger out of the house. (Apparently Alger had been working as a sort of nanny).
"The discovery of the relationship by Emma probably dates to the summer or fall of 1835. Bradley recounted several people who have tried to pin down the date, and noted problems with each date. Some authors have discussed an “embarrassing” incident of polygamy in August 1835. Joseph left for Pontiac, Michigan, possibly to avoid embarrassment for his role. On October 14, 1835, Joseph describes 'dealing with household issues,' possibly a reference to evict Fanny. However, Mark Ashurst-Mcgee suggests this incident refers not to Fanny, but a problem with employees at the printing office.
"Fanny left Kirtland in August or September 1836, so the incident must have occurred prior to that. Bradley notes that dissenters condemned Joseph on July 24, and Joseph left for Salem, Massachusetts, for a treasure trip the next day on July 25.
"Bradley believes Joseph sent Fanny to Missouri at the same time. William McLellin gave his famous quote about having 'no confidence' in Church leadership around this time as well. Fanny soon married non-member Solomon Custer after just a six-week courtship. Bradley believes it may have been a cover of legitimacy if Fanny was indeed pregnant."
("Sunstone 2010--A Feminist Recap," by "Mormon Heretic," 17 August 2010, at: http://mormonmatters.org/2010/08/17/sunstone-2010-a-feminist-recap/
For more information, see Bradley's research on polygamy entitled, "Mormon Polygamy Before Nauvoo?: The Relationship of Joseph Smith and Fanny Alger," in "The Persistence of Polygamy: Joseph Smith and the Origins of Mormon Polygamy," Newell G. Bringhurst and Craig L. Foster, ed. (Independence, Missouri: JohnWhitmerBooks, 2010), at: http://www.amazon.com/Persistence-Polygamy-Joseph-Origins-ebook/dp/B004GNEDIM#reader_B004GNEDIM