Aaaand the family in Gothenburg appears in Berg, W. (1939) Genealogiska anteckningar om Göteborgs-släkter, Ser. 2, vol. 9-10, p. 236. Unpublished manuscript at the County archives at Gothenburg with the following information:
Master book-binder Johan Schenk, b.5/1 1635, d. 21/9 1675. M. 18/10 1663 to Susanna Kuhn, b. 1641, buried 8/8 1690, daughter of the book-binder Thomas Kuhn and Elisabet, and re-married 27/12 1678 to Gottfried Meisner who d. 1695. The family must have come from Husum, at least in 1675 one of the sponsors of Henrik was a Mrs. Magdalena Schenk, m. to Mr. Peter at Husum. [NB. There's a Husum in northern Sweden].
Children:
* Agneta, christened 10 May 1666, d. probably that year.
* Agneta, christened 17 June 1667.
* Thomas, christened 27 January 1669, d. before 1674.
* Johan, christened 5 July 1670, d. 1698, apprentice book-binder.
* Thomas, christened 3 June 1674.
* Henrik, christened 27 July 1675, d. 20 April 1677.
* Agneta, christened 8 October 1671, d. before 10 March 1733, m. 1:stly 12 Sep 1693 to cobbler Christoffer Geppert, b. 1671, d. bef 1726. M. 2:ndly to the leather-worker Lorens Holmberg (was dead bef. 1714?)
* Elisabet, christened 2 January 1673, d. 1707, buried 27 February. M. 4 February 1700 to girdlemaker Jurgen Geppert, b. 1668, d. 1719, buried 30 August, in his first marriage.
From this list of children, the only likely candidate as a father of Henrik Schenk of Kristianstad, would be Thomas, b. 1674 (who could easily have had a child in 1716).