I’ve previously posted about the Proto-Uralic “dental spirants”, and on the problems concerning their reconstruction. These are however far from the only PU segments whose reconstruction involves unsolved difficulties.
The velar nasal *ŋ provides examples of some different types of problems. Thanks to direct retention in a good number of Uralic languages (most consistently in Samic, Mari, Khanty, and most parts of Samoyedic, but also in dialects of Erzya and Udmurt), there can be little doubt that a phonemic velar nasal occurred in Proto-Uralic. However, the historical development of the consonant is speckled with irregularities, and uncertainty on where it is and is not to the reconstructed.
A major part of the puzzle is the Ugric treatment. No single regular reflex can be identified in these languages. Instead the consonant appears to bifurcate, with no clear conditioning. One portion of words excretes a following homorganic plosive: *ŋ > *ŋk, further > Hungarian g. Clear examples include e.g. the word for ‘mouse’: PU *šiŋərə > H egér, Mansi *täŋkər, Khanty *ɬööŋkər; contrast Finnic *hiiri, Mordvinic *šeŋəŕ > šejər ~ ševər, Permic *šɨr, etc.
Another portion however fails to participate in this development. In Khanty, a plain nasal *ŋ remains in such words, while in Mansi and Hungarian, the initial reflex seems to have been a spirant *ɣ (which may later have vocalized). For these words, we might posit a distinct Proto-Ugric *ŋ. This split, although never explained, has been one of the traditional arguments for establishing a distinct Ugric group in the first place.
One front of attack that I think could help make some sense of the Ugric mess is to extend the scope of analysis to the other Uralic languages. I’ve conducted an initial survey on the matter, and one result appears to be that none of the > *ŋk words have secure Samoyedic reflexes. Which may sound uninteresting, but does have interesting consequences: as is the case for most of the other supposedly Ugric phonetic innovations (*s > *ɬ, *ś > *s, *k > *ɣ, etc.), there is here, too, no explicit reason to exclude Samoyedic! That is, *ŋk in these words might as well also be an older, East Uralic feature? At least one unclear comparison, PU *jäŋə ‘ice’ (> Ms *jääŋk, Kh *jööŋk id.) ~ PSmy *jåŋkå ‘hole in ice’ possibly even shows the exact same change (unless this is to be analyzed as a suffixed derivative: *jåŋ-kå < *jAŋə-kA). In all cases where PU *ŋ is reflected as PSmy *ŋ (e.g. *suŋə > *təŋə ‘summer’), the Ugric languages, too, point to “unbroken” *ŋ (here: Ms *tuj, Kh *ɬoŋ id.)
The Permic evidence offers some interesting views as well. More on this particular front of investigation later as the research develops, though — I’ve several new ideas, but the details remain in flux by this point. Ask me again in a couple of years if I have not found the time to return to this subject before then. :)
[Edit 2016-03-13: See now my post Proto-Uralic *ŋx? for one possibility.]
A look closer inward, that is investigating the vocabulary particular to Ugric, reveals further complications as well. The relatively clear split between *ŋ and *ŋk that is present in the oldest Uralic material dissipates here into chaos.  In many examples Hungarian points to *ŋ while Ob-Ugric to *ŋk, or vice versa; sometimes Mansi and Khanty disagree with each other, too. Multiple explanations would be possible in this situation… My own working hypothesis is that “Ugric” is merely a western areal subset of the East Uralic branch, and that much of the vocabulary shared between the three branches is to be analyzed as later diffusions, not as common inheritance. Perhaps the change *ŋ > *ŋk is to be dated as an areal innovation as well — but this remains to be seen.
There is also at least one correspondence where positing original *ŋ seems simply mistaken. This is a decent-sized set of words where Hungarian g corresponds to Ob-Ugric *ɣ. To run a small case study, examples include at least:
- H ág ~ Ms *taɣ ~ Kh *ɬaɣïï ‘branch’
- H fog- ~ Ms *puw- ‘to grasp’
- H nyereg ~ Ms *naɣr ~ East Kh *nööɣər ‘saddle’
- H szaguld- ‘to rush’ ~ Ms *šoom- ~ Kh *saaɣəL- ‘to gallop’, Kh *suuɣəm ‘jump’
- H tegez ~ Ms *täwt ~ Kh *tüüɣət ‘quiver’
First off, note that none of these reflexes shows any direct evidence for an original nasal. Usually at least Khanty retains *ŋ as is, and yet here we get *ɣ instead. There also seem to be no examples of the “next-most regular” correspondence: H g ~ Ms *ɣ ~ Kh *ŋ.
I suspect that these words are to be explained, not from a Proto-Ugric *ŋ that was expanded to *ŋk only in Hungarian, but as relatively recent parallel loans, and that the loan originals featured the voiced stop *g. In Hungarian the option to carry this over directly was available; while in Ob-Ugric, where phonemic voiced stops seem to have remained alien ever since Proto-Uralic, the closest substitute was *ɣ.
Numerous other phonetic irregularities also appear here, some of which could also have resulted from parallel loaning: most prominently Mansi *š in ‘to gallop’; Hungarian ny and Core Mansi *a (but Southern Mansi näwrää!) in ‘saddle’; Khanty *üü in ‘quiver’.
Note moreover how the meanings ‘gallop’, ‘saddle’ and ‘quiver’ are all specialized cultural terminology, and hence these are likely to be loanwords related to the Ugric peoples’ adoption of a steppe nomad lifestyle.
In some cases, loanwords of this layer might have reached Permic, too. As a particularly clear case, Hungarian reg ‘morning’ has been considered related to Permic *rög ‘warm’ — yet, Ob-Ugric #reɣ ‘warm’ seems equally relevant. Equating all three is however not possible if we insist on common inheritance: on one hand, the change *ŋ > *ŋk clearly never extended into Permic, and on the other hand, there is no evidence for a reduction *ŋk > *ɣ in Ob-Ugric. Loaning from Hungarian to Permic could be posited, but if so, why not loaning from “Language(s) G” to Permic directly? The latter scenario seems slightly preferrable in light of another example: Permic *mög ‘riverbend’ ~ Ob-Ugric #mVɣəɬ ‘around, circle’, for which no Hungarian cognates are known. 
Even further evidence for a non-nasal origin for this correpondence can be teased, I believe, out of its occurrence in an apparent consonant cluster:
- H buzog (buzg-) ‘to seethe’ ~ Ms *pëësɣ- ~ Kh *paasəɣ- ‘to drip’
If we were to be consistent with the earlier view, and to reconstruct Proto-Ugric *pësŋ-, this would be the sole example of an obstruent + nasal cluster in the inherited Ugric lexicon!  Additionally, the g/ɣ issue is not the only suspicious feature — no, literally every sound correspondence between Hungarian and Ob-Ugric (z ~ *s, u~ *ëë, b ~ *p) is irregular here. The first of these seems especially telling: if an original *g is assumed, it would be plausible to also assume that the first member of the consonant cluster was not *s, but the likewise voiced *z. This then can be assumed to have been substituted by voiceless *s in Ob-Ugric, but by *z in Hungarian… which would bring us to come one step closer to eliminating the sometimes supposed “sporadic” voicing of medial *s in Hungarian. 
Theories involving unattested substrate languages offer, of course, an easy way to explain whatever one wishes. Does my new explanation actually offer any advantage over the traditional view of leaving the origin of “areal” words open? Certainly, much of this hangs in the air so far, but it should be possible to seek further evidence. Perhaps known Turkic / Mongolic / Iranian loanwords can confirm or deny my supposed substitution pattern *g > Hungarian *g ~ Ob-Ugric *ɣ. Closely combing thru the vocabularies of these families (and, perhaps, Tocharian?) might even be able to turn up cognates for some of these words… though don’t hold your breath.
 I know there’s a handy paper covering this topic out there… but alas, I am failing to relocate it right now. Please drop me a hint in the comments if you have the reference at hand?
 Though there’s moreover Samic *moaŋkē ‘bent’, which seems difficult to integrate into any southeasterly loanword scenario.
 There are examples in the common Ob-Ugric lexicon — e.g. #asma ‘pillow’ — but these seem to be generally derivatives.
 This explanation does raise one question. Hungarian /z/ generally originates from Proto-Hungarian *ð, and a substitution *z > *ð would not seem particularly expected. Should we assume that [z] actually first originated as an allophone of /s/ before voiced stops in loanwords? There seem to be quite a few words in Hungarian with the cluster -zəg : -zg-, and various similar ones such as -zd-, though I do not know if any of these are a part of the language’s oldest loanword layer.