First of all, be sure the student has the desired tone in mind. If he can hear that tone in his mind before he plays and imagine that tone coming from his own bell, it sometimes changes everything. Part of that effect is due to not overthinking the "how" of that beautiful tone.

Secondly, the throat does not actually open and close. The pinched sound you are hearing is caused by the player's lessening the size of the mouth cavity, which functions as a resonance chamber for all wind instruments, including simple instruments like harmonica.

We can feel like we close our throat, but that feeling comes from our raising the back of the tongue and other pinching/straining/clenching in the oral cavity. That is often accompanied by excess mouthpiece pressure.

So . . .

Your student may not relate to your description of what he is doing. Consider these means of encouraging that full and open sound:

"Keep your teeth apart."

"Keep your tongue touching your bottom teeth and flat on the bottom of your mouth."

"Listen for that beautiful sound harder than you work for that beautiful sound."

This student's tone issue is common, as you know. Your addressing it carefully and with love and encouragement will teach both of you more than you can now realize.

Good luck