The Finnic verb root *püütä- (Fi. pyytää, etc.) has two distinct senses: ‘to ask for’ on one hand, ‘to hunt’ on the other. These could plausibly be considered connected, with the former as the original sense, the latter developing as an euphemism. At least the former sense also clearly seems to derive as a loanword from Germanic *beudan- ‘to offer’; most likely relatively late from a form such as Old Swedish biūþa.
A competing etymology also exists: that ‘to hunt’ would be instead a derivative *püü-tä-. This finds immediate support within Finnic from two directions. The first is the existence of what look like parallel derivatives, e.g. Finnish pyynti (? < *püü-ntei) ‘hunt’, Estonian püük (? < *püü-kkV) ‘hunt’. Second is the fact that the sense ‘to ask’ shows a somewhat limited distribution, being found only in a number of the more Scandinavian-influenced varieties: Finnish, Karelian, Estonian and Kukkuzi Votic/Ingrian . The more marginal Ludian and Veps, as well as also both mainstream Ingrian and Votic, only know the sense ‘to hunt’.
Sources such as SSA actually suggest a compromise of sorts between these two approaches; according to this, *püütä- would be across the board an original Proto-Finnic verb meaning ‘to hunt’, and only the meaning ‘to ask’ would have developed by Scandinavian influence. This would allow a much earlier date of contact, though I’m not sure what exact benefits this assumption is supposed to have… Even relatively new Swedish loanwords have relatively often reached Karelian through Finnish, and loanwords homonymous with native vocabulary are by no means an unknown phenomenon.
A derivational etymology of course implies an original shorter root *püü. The meaning of this is not immediately obvious, though. SSA refers to a suggestion that this would be = *püü (Fi. pyy etc.) ‘hazelhen’; hence the verb *püü-tä- would have originally meant specificially ‘to hunt for hazelhen’, only later being generalized to ‘hunt’. On the other hand: Fi. pyynti suggests that the original root was actually a verb, since -nti regularly only forms names of actions (e.g. tuo- ‘to bring’ → tuonti ‘bringing, import’; syö- ‘to eat’ → syönti ‘eating’). I would therefore posit something like *püü- ‘to hunt (intransitive)’, *püü-tä- ‘to hunt (transitive)’.
This so far Finnic-internal reconstruction turns out to have connections in Ugric. A verb root *puŋV- has been known for long, reconstructed on the basis of Hungarian fog- ‘to grasp, to catch’ ~ Mansi *puw- ‘id.’ (the lenition *ŋ > *w in the latter may be regular; there does not seem to be inherited vocabulary in Mansi with *-uŋk-). While an original back vowel *u would be troublesome, there is however a natural explanation. As explored in my previous post, several branches of Uralic show evidence for a backing development of Proto-Uralic *ü in the vicinity of velar consonants. This seems to be the case here as well. Finnic *püü-, as uncovered above, therefore suggests that a better reconstruction will be PU *püŋə-.
This yields all reflexes involved quite regularly. *püŋə- > Hungarian fog- has an exact parallel in *püŋə > fogoly ‘hazelhen’, and there is also the rather similar *piŋə > fog ‘tooth’ (although my previous reservations on not fully understanding the intermediate phonetics of this development still apply). In Mansi, only *ü seems to have been subject to this backing: contrast *päŋk ‘tooth’. *püŋə- > *puw- does not have exact equivalents, but Steinitz’ example of *pükkV-nV > *pukńi ‘navel’ remains a decent parallel. In a small article on the topic,  he also cites Northern Mansi /puki/ ‘belly’ ~ Khanty *pökii ‘bird’s crop’. To me it looks like these could perhaps be from a common root with ‘navel’ (*pükkV-j?). UEW gives instead Finno-Permic cognates pointing to *päkkä, but the irregular vowel correspondence leaves me doubtful. 
The similarity between Finnic *püü ‘hazelhen’ and *püütä- ‘to hunt’ does not have to be accidental, though. It might be worth asking if the derivational relationship has instead been the opposite: if PU *püŋə ‘hazelhen’ had rather been derived from *püŋə- ‘to hunt’? This might be further supportable by how many of the reflexes show later suffixation, e.g. Samic *pëŋkōj; Hungarian fogoly; Moksha /povńä/; Livonian pīki (= Es. püük ‘hunt’, as mentioned above?). Selkup /pee-/ ‘to look for’ : /peekä/ ‘hazelhen’ seems particulary interesting (at least as a semantic parallel — I hesitate to claim that this, together with its other Samoyedic cognates, would derive from *püŋə- at all, since the vowel developments would be highly irregular ). The underived appearence of Finnish pyy, Estonian püü etc. on one hand, Khanty *peŋk on the other, could then end up being a kind of a backformation from earlier compound terms, facilitated by the loss of the bare verbal root.
There is a chronological issue with the Mansi data, though. A form such as /puki/ ‘belly’ clearly cannot be taken back to conventional Proto-Mansi *puki: we would expect the usual development *k > [q] > /χ/ to kick in (compare e.g. *taŋk > /toŋχ/ ‘hoof’). For Northern Mansi in particular, it might be feasible to assume similar relatively late backing as in /puŋk/ ‘tooth’, but this then fails to explain the non-Northern reflexes (e.g. West /püxəń/ ‘navel’).
I also have already earlier argued against the traditional reconstruction of Proto-Mansi *ü. Instead of setting up here a marginal Proto-Mansi *ü after all, which occurred only in the context /p_k/, I have a different suggestion: it will be possible to reconstruct here plain *u for Proto-Mansi — if we assume that the contrast *k : *q had already been phonemicized! While many overviews of the velar backness split in Ugric assume that it was only phonemicized by the development *q > /χ/ (in Northern Mansi, most of Eastern Mansi, all of Northern and Southern Khanty, and in pre-Hungarian), the detailed field records still faithfully and consistently transcribe k͔ = /q/ for most of the other Ob-Ugric varieties as well. Actual reference grammars, as opposed to historically-minded works, often recognize the uvulars and velars as distinct phonemes as well. 
I would thus set up the following develoment:
- Pre-Mansi (“Proto-Ugric”) *ku, *uk > *qu, *uq (> North /χu/, /uχ/)
- Pre-Mansi *kü, *ük > *ku, *uk (> North /ku/, /uk/) (after the lowering of primary PU *ü!)
Later on, then, in Western and Eastern Mansi, a back-development *ku, *uk > /kü, ük/ takes place, completing a kind of “cheshirization cycle”, further cemented by *q > /k/ in a few Western dialects (e.g. Pelymka).
Steinitz’ Geschichte des wogulischen Vokalismus (Berlin, 1955) already lists a few examples that show what I mark here as *ku-, as distinct from *qu-. One is Northern /kurɣ-/ ~ Western /kürr-/ ~ Eastern /körɣ-/ ‘to growl’ < *kurɣ-. Further examples occur in loanwords, such as N /kuľ/ ~ E /köľ/ ‘devil’ (← Komi /kuľ/).
Most such words do not seem to have been attested in Southern Mansi, though. If we followed the usual (and also geographically reasonable) assumption that Southern has been the first dialect area to split away, it seems that “disharmonic” *ku- is in most cases only reconstructible for Core Mansi, not Proto-Mansi proper. In native vocabulary, only the marginal example of *puk- from earlier *pükk- seems to be found.
The most important benefit of this reanalysis, however, is that the marginal contrast *k : *q does not need to be limited to the root type *pükk- > *puk-. It will be possible to explore also other similar contrasts, such as *koo : *qoo (> Core Mansi *kuu : *quu). These seem likely explain a variety of rare or seemingly irregular vowel correspondences between the Mansi dialects: e.g. N /kuur/ : W /küür/ ‘oven’, a loanword from Komi /gor/ ‘id.’ More on this later, though…
 Considered either Ingrianized Votic or Voticized Ingrian, depending on who you ask. I would lean on the second, but the last word on the topic has probably not been said yet. — ‘To ask’ is in here most likely a loan from Ingrian Finnish though, so the question does not matter for today’s purposes.
 Steinitz, Wolfgang. 1956. “Zur ob-ugrischen Vokalgeschichte”. — Ural-Altaische Jahrbücher 28: 241–247.
 There also seem to be compareable words in neighboring families, e.g. Evenki /hiken/ ‘sternum’; /hukēn/ ‘crop’ (the latter with further Tungusic cognates). Since these still show h- < *f- < *p-, any possible connection would have to go quite far back, though.
 Janhunen in Samojedischer Wortschatz fails to reconstruct a single PSmy proto-form, giving instead three variants: *pü- (Nenets), *pö- (Nganasan), *pä- (others) (in his reconstruction: *pe-).
 For example, a phonemic contrast /k/ : /q/ is explicitly presented for Surgut Khanty in Márta Csepregi’s recent reference grammar Szurguti osztják chrestomathia (Szeged, 1998).