I was recommended another source of Hungarian audio:
I asked my friends and they recommended Klub rádió as a station which broadcasts mostly speeches, interviews and conversation. You can listen to it online here:
On this page you will also have several other radio stations to choose from.
I don't know how to acquire the stream endpoint, but there is an option to record a fragment of the broadcast in mp3. On the top right corner of the page you see the sign "Rádióadás rögzítése" with a little microphone icon. Under the sign you can set the length of the recording in minutes from 1 to 30. Once the length is set, click the link "Rögzítés megkezdése" below. It will immediately prompt you to save the mp3, then it starts saving the fragment into the mp3 file you created.
I was recommended another source of Hungarian audio: