Hún segir að stuttur göngutúr og heitt kaffi hjálpi henni þegar hún er syfjuð og þarf að teygja sig.

Here að introduces a content clause, the equivalent of English that.

So:

Hún segir að ...
= She says that ...

In English, that is often omitted, but in Icelandic að is very common and standard in this kind of sentence.

Hjálpi is the present subjunctive form of hjálpa.

Icelandic often uses the subjunctive in subordinate clauses after verbs like segja, halda, telja, and similar verbs of saying or thinking, especially when the clause is presented as reported information rather than a plain fact directly asserted by the speaker.

So:

• hjálpi = subjunctive
• hjálpar / hjálpa = indicative

Very roughly, the subjunctive here gives a reported or less directly asserted feel: she says that a short walk and hot coffee help her.

In meaning, the subject is plural: stuttur göngutúr og heitt kaffi = a short walk and hot coffee.

But the form hjálpi does not show the difference between 3rd person singular and 3rd person plural in the present subjunctive. They look the same.

So in this sentence, you understand the number from the subject, not from the verb ending.

If this were in the indicative, the plural would be clearer:

• hjálpar = singular
• hjálpa = plural

Because the verb hjálpa takes the dative case, not the accusative.

So:

• hjálpa henni = help her
• not hjálpa hana

This is something English speakers often just have to memorize with the verb:

• hjálpa einhverjum = help someone
• einhverjum is dative

Because Icelandic adjectives agree with the noun in gender, number, and case.

Here:

• göngutúr is masculine singular nominative
  → stuttur göngutúr
• kaffi is neuter singular nominative
  → heitt kaffi

So the adjectives change to match the nouns they describe.

This is also why the two adjectives do not look the same even though both are translated with simple English forms like short and hot.

Because syfjuð agrees with hún, which is feminine singular.

The adjective is basically:

• masculine: syfjaður
• feminine: syfjuð
• neuter: syfjað

Since the subject is hún = she, the feminine form is used:

• hún er syfjuð = she is sleepy

Þegar hún er syfjuð is a new subordinate clause, so it normally has its own explicit subject: hún.

Inside that clause, there are then two coordinated verbs:

• er syfjuð
• þarf að teygja sig

Since both refer to the same subject, Icelandic does not need to repeat hún before þarf.

So the structure is basically:

• when she is sleepy and needs to stretch

not

• when she is sleepy and she needs to stretch

The second she can be omitted because the subject is already clear.

Because þurfa is commonly followed by að plus an infinitive.

So:

• þarf að teygja sig = needs to stretch

This is a very common pattern:

• þarf að fara = needs to go
• þarf að læra = needs to study
• þarf að hvíla sig = needs to rest

Because sig is the reflexive pronoun, used when the action goes back to the subject.

So:

• hún þarf að teygja sig = she needs to stretch herself
• hún þarf að teygja hana would mean she needs to stretch her, referring to some other female person

English usually just says stretch, but Icelandic often uses a reflexive construction here: teygja sig.

Icelandic does not have an indefinite article like English a/an.

So:

• stuttur göngutúr can mean a short walk
• heitt kaffi can mean hot coffee or a hot coffee, depending on context

That is completely normal. Icelandic often leaves nouns indefinite without adding any extra word.