Translation:The students have difficulty hearing the professor.
If I took out the "med" here, would the sentence be conveying the same meaning?
Is "med" usage idiomatic? I do not get when to use it. For example which one is right? "jeg kan ikke vente med å prøve" or "jeg kan ikke vente å prøve"
I have been wondering the same and checked a few pages. It seems that in Norwegian "å vente" can mean: wait, await, anticipate. There are several ways in which it connects with other words in the sentence.
You can have no "follow up", so sentences such: I am waiting (Jeg venter), Can you wait here a few minutes? (Kan du vente her noen minutter?), Wait and see (Vent og se)
You can have a noun after and then you need to use "på" after, such as: I am waiting for you (Jeg venter på deg), They are waiting for the ambulance (De venter på ambulansen)
You can have a verb after and then you need to use "med": Jeg venter med å rydde til morgen, Jeg kan ikke vente med å snakke med ham.
There are more complicated use cases that I found such as: Middagen venter! (The dinner is ready, it's "waiting"), You can expect ("anticipate") me at 6 (Du kan vente meg klokka seks)
I'm not Norwegian so it would be good to have some confirmation from a native :)
It looks good to me! :)
In the sense of anticipating/expecting it can also be used with a noun after it without a "på", but you're correct in saying that it would require a "på" if the meaning is "to wait".
General/Impersonal statements about anticipation/expectation are often prefaced by "Det ventes...", which could translate to "It's expected to/We're expecting/You can expect.." depending on the context:
"Det ventes regn (/å regne) i morgen."
"It's expected to rain tomorrow."