Learn Dutch: Hij hoest met zijn mond dicht. - He coughs with his mouth closed.

The current lesson is about using wat to mean 'somewhat' or 'a little'. Where would I put wat in this sentence to say 'He coughs a little with his mouth closed'?

You would place it right after the verb: Hij hoest wat met zijn mond dicht. In this position, the adverb wat softens the action, meaning he is doing just a little bit of coughing.

Why use dicht for 'closed' here instead of gesloten?

For physical things that open and shut—like a mouth, a door, or a window—dicht is the most natural and common choice in everyday Dutch. Gesloten is technically correct but sounds much more formal or clinical.

Is 'met zijn mond dicht' the standard way to describe doing things with your body in Dutch?

Yes! Just like in English, Dutch uses met (with) + [body part] + [state or position] to describe how an action is performed. You will see this exact same structure in common phrases like met je ogen dicht (with your eyes closed) or met je handen in je zakken (with your hands in your pockets).

Why don't we put the adjective before the noun and say met zijn dichte mond?

While met zijn dichte mond (with his closed mouth) is grammatically possible, it sounds very unnatural. To describe the temporary state of a body part while doing something else, Dutch strongly prefers putting the state at the end of the phrase (met zijn mond dicht).