[This is part of a series of defectivity case studies.]

Hungarian verbs exhibit one of better-documented cases of defectivity, discussed first in English by Hetzron (1975). In particular, the study by Rebrus & Törkenczy (2009), henceforth RT, is a wealth of information, though this information is presented in so many different forms that it has taken me some effort to extract the basic generalizations. Furthermore, they seem to have deliberately avoided putting forth an actual analysis. Rather than trying to explain the data RT’s way I’ve tried to put matters into my own words. Note that below I often give multiple allomorphs for a suffix or epenthetic segment; these are conditioned by Hungarian’s well-known process of vowel harmony (see, e.g., Siptár & Törkenczy 2000 for description).

Hungarian has a class of verbs, mostly intransitive, which form a 3sg. indefinite present indicative in -ik rather than the ordinary null ending. Of these, some such verb stems end in surface consonant clusters. When consonant-initial suffixes (e.g., the imperative -j, the imperative subjunctive -d, the potential -hat/-het) are added to verb stems of this form, a fleeting vowel (o, ӧ, or e) normal breaks up the stem cluster (e.g., ugrik ‘s/he jumps’,  ugorhat ‘s/he may jump’). It is not clear to me whether this fleeting vowel is underlying or epenthetic, though it seems to be standard in Hungarian philology to assume the latter. However, for certain verbs, this does not occur and the verb is simply defective (e.g., csuklik ‘s/he hiccups’ but *csuklhat, *csukolhat). According to Hetzron  (ibid., 864f.)  all the defective verbs have stems of the form /…Cl/ or /…Cz/, though it is not the case that all verbs of this form are defective (e.g., vérzik ‘s/he bleeds’, vérezhet ‘s/he may bleed’; hajlik ‘s/he bends’, hajolhat ‘s/he may bend’). A study by Lukács, Rebrus, and Törkenczy (2010), conducted detailed grammaticality judgment tasks and appears to largely have confirmed the description given by Hetzron and RT.

What is the source of these patterns?  It may be that whatever causes the fleeting vowel to be epenthesized is less than fully productive; but then it is also necessary to also appeal to absolute phonotactic illformedness to derive the defectivity. It is not obvious that this will work either, because at least some of these clusters ought to simplify according to RT.

References

Hetzron, R. 1975. Where the grammar fails. Language 51: 859-872.
Lukács, Á., Rebrus, P., and Törkenczy, M. 2010. Defective verbal paradigms in Hungarian: description and experimental study. In M. Baerman, G. C. Corbett, and D. Brown (ed.), Defective Paradigms: Missing Forms and What They Tell Us, pages 85-102. Oxford University Press.
Rebrus, P., and Törkenczy, M. 2009. Covert and overt defectiveness in paradigms. In C. Rice and S. Blaho (ed)., When Nothing Wins: Modeling Ungrammaticality in OT, pages 195-234. Equinox.
Siptár, P., and Törkenczy, M. 2000. The Phonology of Hungarian. Oxford University Press.