What does "gjerne" add to the sentence that "Jeg vil ha en kopp kaffe" does not get across?
It makes it more polite, like "I want a cup of coffee" vs. "I would like a cup of coffee."
Gjerne is basically exactly the same as 'gern' in German. Makes Things easier to understand for speakers of German.
If I'm not mistaken, gjerne is an adverb which translates to gladly. Thus, he/she is saying "I will gladly have a cup of coffee.
no, I wouldn't translate that to gladly in this sentence. Gjerne is used commonly in the German language as well "Gerne". If you'd order something in a restaurant you would say: "Ich hätte gerne eine Tasse Kaffee", so "would like" is the phrase I would take here.
And "graag" in Dutch. The word means "gladly," or "with pleasure," but has common usage like this. The sentence, which probably was in a Dutch lesson here, would be "Ik wil graag een kop koffie."
So would a Norwegian be more likely to say 'jeg vil ha' or 'jeg vil gjerne ha' if they were at a restaurant for example? Which is the most common?
Could 'gjerne' be cognate with 'yearn', though maybe only in a long-ago English?
'Gjerne' comes from the Old Norse word 'gjarna' meaning willingly, which comes from the Proto-Germanic word 'gernô' which is the adverb of the adjective word 'gernaz' also meaning willingly or eager.
'Yearn' comes from the Middle English word 'yernen' which comes from the Old English word 'giernen' which itself comes from Proto-Germanic 'girnijaną' meaning to want or desire. 'Girnijaną' is a verb that's derived from the adjective 'gernaz.'
TLDR: Yes, they come from the same Proto-Germanic roots.