Þegar kemur að heimaþjónustu er hjálpartækið niðurgreitt eftir þarfamat.

In this sentence, þegar kemur að is an idiomatic expression meaning when it comes to, as for, or regarding.

Even though þegar often means when, this whole chunk is best learned as a set phrase:

• þegar kemur að X = when it comes to X

So:

• Þegar kemur að heimaþjónustu = When it comes to home care / home assistance

It does not literally mean that something is physically coming somewhere in this context.

Kemur is the 3rd person singular present form of koma (to come).

In the expression þegar kemur að ..., Icelandic uses this fixed singular form in an impersonal way, much like English says when it comes to ... with it.

So you do not need to look for a visible subject here. Just treat:

• kemur að = part of the idiom

This is very common in Icelandic formal and semi-formal language.

Because the preposition að takes the dative case.

The base noun is:

• heimaþjónusta = home service / home care

Its dative singular form is:

• heimaþjónustu

So:

• að heimaþjónustu = regarding home care

This is one of the most important things to watch in Icelandic: prepositions often control a specific case.

Heimaþjónusta is a compound noun:

• heima = at home
• þjónusta = service

Together, it means something like:

• home service
• home assistance
• home care services

The exact English translation depends on context. In public-service or healthcare language, it often refers to support provided to people in their homes.

This is because of the V2 rule in Icelandic.

In main clauses, the finite verb usually comes in the second position. Here, the sentence begins with a fronted adverbial/subordinate element:

• Þegar kemur að heimaþjónustu = first slot

That means the finite verb of the main clause comes next:

• er

Then the subject follows:

• hjálpartækið

So the structure is roughly:

• [Þegar kemur að heimaþjónustu] [er] [hjálpartækið] [niðurgreitt] [eftir þarfamat].

This is very normal Icelandic word order.

The basic noun is:

• hjálpartæki = assistive device, aid, helping device

It is a neuter noun.

The ending -ið is the definite article attached to the noun, so:

• hjálpartæki = an assistive device / assistive devices
• hjálpartækið = the assistive device

Icelandic usually attaches the to the end of the noun instead of using a separate word like English does.

Niðurgreitt means subsidized.

It comes from the verb niðurgreiða, meaning to subsidize or to pay part of the cost.

Here it is used like a past participle/adjectival form with er:

• er niðurgreitt = is subsidized

The ending -tt appears because it agrees with hjálpartækið, which is:

• singular
• neuter
• definite

So the form is:

• niðurgreiddur = masculine
• niðurgreidd = feminine
• niðurgreitt = neuter

Since hjálpartækið is neuter, niðurgreitt is the correct form.

Yes, functionally it works much like an English passive or passive-like description:

• hjálpartækið er niðurgreitt = the assistive device is subsidized

In Icelandic, vera + past participle/adjectival form is often used to describe a state or resulting condition. In many contexts, the best English translation is a passive.

So for a learner, it is perfectly reasonable to understand this as:

• is subsidized

Eftir þarfamat means according to a needs assessment or following a needs assessment.

The words are:

• eftir = after, but also often according to / based on
• þarfamat = needs assessment

In administrative or healthcare language, eftir often has this sense of on the basis of.

So here it means the subsidy is given based on an evaluation of the person's needs.

Because Icelandic often leaves nouns indefinite in general, procedural, or institutional statements where English might also say a or use no article.

So:

• eftir þarfamat = according to needs assessment / according to a needs assessment

In natural English, we usually say:

• after a needs assessment
• according to a needs assessment

The Icelandic form without the definite article sounds normal here because it refers to the process or basis of assessment, not one specifically identified assessment already known to the listener.

Yes. Icelandic forms compounds very freely.

Þarfamat is made from:

• þörf = need
• mat = assessment, evaluation, appraisal

Together:

• þarfamat = needs assessment

This is very typical Icelandic vocabulary-building. Learning to spot compounds will help a lot with reading.

A more literal breakdown would be:

• Þegar kemur að heimaþjónustu = When it comes to home service/home care
• er hjálpartækið niðurgreitt = is the assistive device subsidized
• eftir þarfamat = according to a needs assessment

So the literal structure is close to:

• When it comes to home care, the assistive device is subsidized according to a needs assessment.

This kind of literal unpacking is useful, but it is better to remember some parts as chunks:

• þegar kemur að = when it comes to
• er niðurgreitt = is subsidized
• eftir þarfamat = according to a needs assessment

Yes, a few common ones:

• Þ in Þegar is like th in thing
• ð in words like niðurgreitt is like th in this in many contexts, though actual pronunciation can vary
• hj in hjálpartækið is not pronounced like English hj; it has a more breathy/palatal quality
• æ in tækið is pronounced somewhat like eye
• ll and nn patterns in Icelandic can also behave differently from what English speakers expect, though they are not the main issue in this sentence

A rough learner-friendly approximation might be:

• THYEG-ar KEM-ur ath HAY-ma-thyon-ustu er HYAUL-par-tye-kith NITH-ur-graytt EFF-tir THARF-a-mat

That is only approximate, but it can help as a first step.