You can guess on its existence by hearing hallgatja and knowing that it's the definite conjugation, which needs a definite object. Which in most cases is indicated by the article a.
Me too - I realized it was definite so kept listening - then assumed I'd made a mistake with the verb.
I did hear: Peter hallgatja tanart', the 'a' was not audible at all.
No, you can't translate this so directly. The grammar here is very different and the 'to' wouldn't be needed.
The suffix -gat/-get is a ... let's call it 'frequentator'. It usually indicates if something is done several times and/or with more intensity. For instance, you have beszél - to speak, and beszélget - to talk with one another. Or néz - to look, and nézeget - to look around or inspect. And just like this, the suffix makes hall - to hear - to a more instense hearing, with the meaning of 'to listen'.
Thank you for the explanation. Are you a native speaker of Hungarian, or do you know so much because you are already so advanced?
I am not a native, but I have busied myself for a while with the grammar of the language.
Thanks for the explanation about frequenters, it really helps to clear this up.