Jeg vil gerne have endnu en kop kaffe.

Literally, Jeg vil gerne have is something like I want gladly to have, but in natural English it usually means I would like or I’d like.

This is a very common Danish way to make a request sound softer and more polite than just Jeg vil have.

• jeg = I
• vil = want / will
• gerne = gladly / willingly / with pleasure
• have = have

So Jeg vil gerne have ... is a standard polite pattern for ordering or asking for something.

Gerne makes the sentence sound more friendly and less direct.

Compare:

• Jeg vil have en kop kaffe = I want a cup of coffee
  This can sound a bit blunt.
• Jeg vil gerne have en kop kaffe = I would like a cup of coffee
  This sounds more polite and natural.

A learner can think of gerne here as helping turn want into something closer to would like.

Because Danish often uses a modal verb plus an infinitive, just like English does.

Here:

• vil is the finite verb
• have is the infinitive

So structurally it works like:

• I want to have ...
• Danish: Jeg vil have ...

When gerne is added, it becomes the polite expression Jeg vil gerne have ...

Endnu en means another or one more.

So:

• en kop kaffe = a cup of coffee
• endnu en kop kaffe = another cup of coffee / one more cup of coffee

In this sentence, the speaker already has had coffee, or at least is asking for an additional cup.

Because endnu en usually means one more / an additional one, while en anden usually means a different one / another one of a different kind.

Compare:

• Jeg vil gerne have endnu en kop kaffe
  = I’d like another cup of coffee.
  Meaning: one more cup.

• Jeg vil gerne have en anden kop kaffe
  = I’d like a different cup of coffee.
  Meaning: not this one; another one instead.

That difference is very important.

In Danish, with many measure expressions, you normally say them directly without af:

• en kop kaffe = a cup of coffee
• et glas vand = a glass of water
• en flaske vin = a bottle of wine

This is very natural Danish structure. Using af here would usually sound wrong or unnatural.

So kop kaffe works like a unit: cup + coffee.

Because kaffe is an uncountable noun here, just like coffee in English.

You say:

• en kop kaffe = a cup of coffee

not:

• en kop en kaffe

The article belongs to kop, because kop is the countable noun. Kaffe just tells you what kind of cup it is: a cup of coffee.

En is the common-gender indefinite article, equivalent to a / an.

The noun kop is a common-gender noun:

• en kop = a cup

So in endnu en kop kaffe, the en belongs to kop.

If the noun were neuter, you would use et instead:

• endnu et glas vand = another glass of water

This is standard Danish main-clause word order:

• Jeg = subject
• vil = finite verb in second position
• gerne = adverb
• have = infinitive
• then the object: endnu en kop kaffe

So the sentence follows a very normal Danish pattern: subject + finite verb + adverb + infinitive + object

That is why gerne comes after vil and before have.

Yes, it is polite, normal, and very natural.

You could say it in a café, at someone’s home, or in many everyday situations. It is a perfectly good way to ask for more coffee.

That said, Danes also often use other common request patterns, for example:

• Kan jeg få endnu en kop kaffe? = Can I have another cup of coffee?
• Må jeg bede om endnu en kop kaffe? = May I ask for another cup of coffee?

These can sound a bit more like direct request formulas, especially in service situations.

In careful pronunciation, it is roughly:

yai vِل GERN-e HA-ve EN-no en kop KAF-e

But in real speech, several parts are reduced:

• jeg often sounds more like yai
• vil gerne can become quite smooth and compressed
• have may sound more like ha
• kaffe is usually two syllables: KAF-e

A rough learner-friendly version might be:

yai vil ger-ne ha en-no en kob kaf-e

The exact sounds vary, but the biggest thing to notice is that spoken Danish is often more reduced than the spelling suggests.

Yes, and it is very close in meaning.

• endnu en kop kaffe = another cup of coffee / one more cup of coffee
• en kop kaffe mere = one more cup of coffee

Both are natural.
A small difference:

• endnu en focuses on another one
• mere focuses on more

In practice, both can work well in this situation.

Because in a normal Danish main clause, the finite verb usually comes in second position.

So:

• Jeg vil gerne have ... is correct.

But jeg gerne vil have can appear inside a subordinate clause, where the word order changes. For example:

• Han siger, at jeg gerne vil have endnu en kop kaffe.
  = He says that I would like another cup of coffee.

So the sentence changes depending on whether it is a main clause or part of a subordinate clause.

Not literally by itself, but the whole expression often corresponds to English would like.

On its own, vil usually means want or will. But in the pattern vil gerne have, the natural English translation is often more polite:

• Jeg vil gerne have ... = I would like ...

So this is a good example where you should translate the whole phrase, not each word separately.