Höfuðverkurinn minnkar þegar ég drekk heitt te.

Because it is built as a compound plus the definite article:

• höfuð = head
• verkur = pain, ache
• höfuðverkur = headache
• -inn = the suffixed definite article, roughly the

So höfuðverkurinn literally breaks down as head-ache-the = the headache.

This kind of long compound is very normal in Icelandic.

The ending -inn is the definite article attached to the noun. Icelandic usually puts the at the end of the noun instead of using a separate word.

So:

• höfuðverkur = a headache / headache
• höfuðverkurinn = the headache

In this sentence, it refers to a specific headache, not just headaches in general.

It is in the nominative singular because it is the subject of the verb minnkar.

The structure is:

• Höfuðverkurinn = the subject
• minnkar = decreases / gets less

So the thing doing the decreasing is the headache, which is why the noun is nominative.

Minnkar is the 3rd person singular present tense of minnka.

Here it means something like:

• decreases
• lessens
• gets smaller
• in natural English here, often gets better or eases

Because the subject is höfuðverkurinn (a singular third-person noun), the verb appears as minnkar.

Because drekk is the correct 1st person singular present form of the verb drekka = to drink.

So:

• að drekka = to drink
• ég drekk = I drink
• þú drekkur = you drink
• hann/hún/það drekkur = he/she/it drinks

This is a very common pattern in Icelandic: the dictionary form is the infinitive, but the actual sentence uses a conjugated form.

Because the adjective has to agree with the noun.

• te is neuter singular
• the object of drekk is in the accusative
• the adjective heitur = hot must match that noun

So the correct form is:

• heitt te = hot tea

Here heitt is the neuter singular form of heitur. In this case, the neuter singular accusative looks the same as the nominative.

Because heitt te is indefinite: hot tea, not the hot tea.

Compare:

• heitt te = hot tea
• heita teið = the hot tea

When the noun is definite, Icelandic usually uses:

1. the suffixed definite article on the noun, and
2. the weak form of the adjective

That is why heitt changes to heita when you say the hot tea.

Þegar means when and introduces a subordinate clause.

So:

• Höfuðverkurinn minnkar = the headache decreases
• þegar ég drekk heitt te = when I drink hot tea

Together, the second part tells you when the first thing happens.

Yes. After þegar, the subordinate clause has normal order:

• þegar ég drekk heitt te
• when I drink hot tea

That is:

• þegar = when
• ég = I
• drekk = drink
• heitt te = hot tea

This is completely normal Icelandic word order for a subordinate clause.

Yes. You can also say:

Þegar ég drekk heitt te, minnkar höfuðverkurinn.

That is also correct.

A useful thing to notice is that when the subordinate clause comes first, the finite verb in the main clause usually comes next:

• Þegar ég drekk heitt te, minnkar höfuðverkurinn.

Not:

• Þegar ég drekk heitt te, höfuðverkurinn minnkar.

This is because Icelandic is basically a verb-second language in main clauses.

Because the sentence expresses a general or repeated situation, not just one event happening right now.

So the present tense here means something like:

• Whenever I drink hot tea, the headache gets better
• My headache tends to lessen when I drink hot tea

Icelandic often uses the present tense this way, just like English does in sentences about habits or general truths.

Literally, it means the headache.

However, depending on context, English might naturally translate it as my headache if it is obvious that the speaker is talking about their own headache.

If you wanted to make my fully explicit in Icelandic, you could say:

• Höfuðverkurinn minn minnkar þegar ég drekk heitt te.

But the original sentence does not explicitly contain minn; that idea would come from context, not from the grammar alone.