The family appears in Iván Nagy's (apologies for the reversal of the names, Hungarian custom is to put the last name first, but for clarity here I'm using a western European order with the personal name first and the surname second). Magyarország családai, which is to say "Hungarian Families", vol. 11. This is an early book, however, and although your Franz has been born, he is too young at the time of publication to have been married or had offspring.
Unfortunately I have absolutely zero Hungarian and cannot translate the details about this family. However, do note that "af Wenckheim" is inaccurate ("af" is a Scandinavian particle, whereas "von" would be more appropriate here).
A quick google also reveals plenty of information about the family, the majority of which is Hungarian, so I think you're spot on with the Austro-Hungarian Empire.
Apparently there were two branches of the family, a baronial and a comital branch. Franz great-grandfather was made count in 1802.
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A second source, the journal Turul, has an article in the 1904 issue, pt. 1, called "A SPANYOL KIRÁLYOK MAGYAR VÉREI". Ferencz (Franz) is listed here as born 1855, and married first 1887 to baroness Klára Radák (1866-1889); he married secondly, 1893, to countess Ilona Forgách (born 1871). I suppose that they must have divorced or that she died shortly after this journal was published given that he married again in 1905.
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Siebmacher's Wappenbuch, "Der Adel von Ungarn" notes that August Wenck was ennobled by Emperor Rudolf II on the 16th of January 1608, with the name "von Wenckheim". The brothers József and Xavier-Ferencz were raised to barons on the 18th of December 1776 (apparently as "Wenck von Wenckheim"). József was then raised again, along with his son József, on the 9 April 1802 to become counts (gróf/graf von Wenckheim).
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Apologies for the poor referencing here, but I hope it's enough to go on. Here's the pedigree from Nágy Iván, Magyarország családai, vol. 11.