In the UEW we find a rough Proto-Ugric reconstruction *pukkɜ ‘blunt end of a tool’, with divergent later semantic development: ‘eye of needle’ in Ob-Ugric, ‘back of hammer/ax/knife/…’ in Hungarian fok. There is reason to suspect though that if related, these words do not go back to a simple bisyllabic root. Mansi *pup could be maybe in principle derived from *puK-p. However, *ɣ in Khanty *poɣ does not correspond to Hungarian k! The normal Khanty reflex of *kk is unlenited *k.  This discrepancy clearly shows UEW’s reconstruction to be overly impressionistic. Still, the comparison as such does not have to be abandoned: it can be instead approached as a family of three different derivatives, *puCV-ka, *puCV-pV, *puCV-kka with some lost weak medial consonant.
The identity of this lost consonant has been discovered by now, too. While rooting around for references on Samoyedic etymology, I have found that Helimski in an apparently little-known 2001 paper, “PU *i̮ś- ‘to cause to be, to be’ and some other core vocabulary items in Proto-Uralic”,  in passing connects to the Ugric words with a newly set-up Samoyedic *puj ‘blunt end of a tool (eye of needle, back end of sled runner)’. Loss of *-jə- in derivatives is regular in Ob-Ugric; in Hungarian the conditioning might be rather *uj > *u or *jkk > *kk > *k. The meaning ‘eye of needle’ as a Siberian Uralic semantic isogloss is interesting; maybe it is not a common innovation, but rather an archaism that has not survived in Hungarian.
Helimski however does not seem to have noticed that this new reconstruction allows a few further etymological connections too. At least Mansi *puj, Khanty *puuj ‘back part’ is obviously connectable as continuing the underived basic root of all the ‘blunt end’ reflexes. (The Khanty vocalism is, once again, difficult to explain though; maybe it’s a loan from Mansi.) UEW (s.v. *pujɜ) also gives several other generic spatial reflexes from Samoyedic that go back to *puə. Northern Finnic *poo ‘butt’ is often also considered a reflex, but this runs into phonological problems. My expectation would be for *pujə to yield instead *pui or at most *puu.
Given the new evidence of reflexes referring to tools, I would suggest that better Finnic cognates can be found, showing phonological development as expected. For one, there’s the word family including Finnish puikko ‘(narrow) stick, rod’, puikkari ‘net needle’, puikkaa- ‘to stick (in)’, earlier probably something like *’to poke with a blunt tool’. These can be taken as derivatives from a (pre-)PF stem *puikka. It would seem to be an exact equivalent of Hu. fok < *pujə-kka, but different meanings suggest they more likely have been formed independently. A different direction of derivation appears in the adjectives puikea ~ pujea ‘oblong = having a defined end’ < *puj-(k)əta. Furthermore pujo ‘narrow; narrow object’ could belong here (probably as a late derivative within Finnish, something like ancient *pujə-w I’d expect to give **puju). — SSA mentions a different etymology for the *puikka family: derivation from puu ‘tree, wood’ (or rather, from the plural stem pui-), but the supposed semantics in this seem too vague.
Now that there is much more backing across Uralic available, even the old comparison by Setälä of pujo with Samoyedic *pujå ‘nose’ could be rehabilitated, now on the root level: the latter seems to be analyzable as continuing something like *pujə-ja or *pujə-la, roughly ‘pointed thing’ (a relatively typical origin for terms for ‘nose’). The exact details of morphology will have to remain a bit up in air for now though… normally *-ja derives agent nouns, *-la local nouns.
Some phonological considerations on the development of *-Vjə rimes in Samoyedic will be also required, since the divergence between *puj ‘eye of needle, etc.’ and *puə ‘back part’ should be explained somehow. For now I will leave this to a few notes. On one hand, actually many reflexes such as Tundra Nenets /pū/, Mator hu-na- could still derive from either proto-form. It’s maybe conceivable that the two PSmy reconstructions could be just positional variants of each other; e.g. *puj as a self-standing noun vs. *puə- before further suffixes? But on the other hand, at least Nganasan /hüj/ ‘eye of needle, back end’ vs. /huə/ ‘back part’ are difficult to treat in this way. For now it seems more feasible to suggest that, despite looking like an underived basic root noun, the semantically derived *puj actually also goes back to some kind of a derived pre-form; options that would work without too many new assumptions could include e.g. *pujə-ka (akin to Khanty), *pujə-k, *pujə-j.
 The longest-known example is probably Kh. *ɭökəmə- ‘to push’ ~ Hu. lök-, Fi. lykkää-.
 Published in the workshop proceedings collection Budapesti Uráli Műhely II. This is based on (and covers some, though not all, of the same ground as) the unpublished presentation “Basic Vocabulary in PU and PFU: Remarks to Etymology and Reconstruction” that I’ve seen cited in a few places.