I notice that the sources for the connection between Tzvi Heller Kahana, of Polonoye and the parents Yaakov Yosef Kahana are from MyHeritage Trees.
In Otto Muneles description of the gravestone of Raizel Segal Kahana, [d.#1 Tosfot Yom Tov] he says that two of her sons who are also on the same gravestone, Moshe and Simcha, died with the mother during the plague of 1639 in Prague, and that the mother was survived by a son, Michal. (Muneles, Ancient Jewish Cemetery of Prague, p. 323). Muneles does not mention either Nachman Kahana Kahana-Heller or a Tzvi Heller Kahana, of Polonoye, or a Rabbi Judah Kahana Heller, of Kalish However, I believe the latter is the son of the Michal mentioned by Muneles (i.e. the grandson of the deceased woman Raizel through her surviving son Michal), so I leave him out of the following paragraph.
It is possible that Munele did not know about these two sons (which seems unlikely), or that the two are sons of the father Yaakov Yosef Kahana from a second marriage (after his first wife died in the 1639 plague), but I see no evidence for this option. Can someone point out any evidence for the latter option, if known?
In the absence of such evidence I would propose that the two sons be detached from both Yaakov Yosef Kahana and that the third son, Rabbi Judah Kahana Heller, of Kalish be re-connected to a new father: Michal, son of Yaakov Yosef Kahana and Raizel Segal Kahana, [d.#1 Tosfot Yom Tov]. The proposal regarding the son Judah is supported by the blogpost cited in a separate Discussion within the profile of Yaakov Yosef Kahana.
Thanks for the attention to this matter.
Private User Neil Rosenstein also has Rabbi Judah Kahana Heller, of Kalish as the son of Yaakov Yosef Kahana, in the family trees in the back of The Fast and the Feast. Of course there is no source referenced, and I think there are other problems with his version of the Kahane tree.