I'm not sure where to start with Hungarian punk. What I can give is some impressions, little trivia and some insight into it, I don't want to turn to Wikipee to balance out my lack of knowledge.

Maybe I would start with the most successful punk band which was criticized being a "sellout" and such, as it's usually done. Their music turned more heavy during the years, so they aren't purely punk. This I listened the most, know the best, but I also listened very little from their late albums, after 2002, but from before that there are great many songs I'm fond of, even tho I can't listen most of them due to nostalgic reasons...
This band is the Tankcsapda (= Czech hedgehog, anti-tank obstacle).
They started out in 1989. Their records are fairly quality sounding, the music is simple (which is given in punk rock), lyrics are often well written, frequently in a playful fashion (in meaning, or with the rhythm and rhymes). Topics in most cases are centered around "sex, drugs and rock n roll", with little politics here and there, probably this is why they gained more popularity, their stuff was more easily digestible for a wider audience.

I already posted a song of them (Akapulkó) in previous music threads, now another couple:
1. Mitől legyen? - (~ Why should I?) a classic from the first album.
https://youtube.com/watch?v=lg8Cvkkzkmc [Embed]
I give a translation (not an artistic one) for this one.
I'm here, singing
What can I do beside
To have more on my pocket
But why should I be happy?

I'm standing, looking at you, I don't understand you
If you knew how I suffer