Geturðu kastað jakkanum mínum hingað?

It is basically getur + þú:

• getur = can / is able to
• þú = you

So Geturðu ... ? means Can you ... ?

In Icelandic yes/no questions, the verb often comes first, and þú is very commonly reduced and attached in writing as -ðu. So:

• Getur þú ... ? = full form
• Geturðu ... ? = very common contracted form

Both mean the same thing.

After the modal verb geta (can / be able to), Icelandic uses the supine form of the main verb. For kasta, that form is kastað.

So:

• geta kastað = can throw
• geturðu kastað = can you throw

This form often looks like a past participle to English speakers, but here it is not past tense. It is just the form Icelandic uses after geta.

Compare:

• Ég get komið. = I can come.
• Hún getur séð þetta. = She can see this.

Because after modal verbs like geta, Icelandic does not use að before the following verb.

So you say:

• Geturðu kastað ... ?

not:

• Geturðu að kastað ... ?

This is similar to English can throw, not can to throw.

Because the verb kasta takes its object in the dative case.

So Icelandic is not using the basic dictionary form jakki here. It changes the noun to the form required by kasta.

Very roughly:

• jakki = jacket (dictionary form)
• jakkanum = the jacket in dative singular

So kasta jakkanum means throw the jacket, with the case that kasta requires.

This is something you will see a lot in Icelandic: certain verbs require specific cases.

Because Icelandic very often puts the possessive after the noun, and when it does, the noun is usually definite.

So:

• jakkanum mínum = my jacket
  literally something like the jacket of mine

Also, the possessive must match the noun in gender, number, and case.

Here:

• jakkanum is masculine, singular, dative
• so minn becomes mínum

That is why you get:

• jakkanum mínum

not:

• minn jakki in this sentence

A fronted possessive like minn jakki is possible in Icelandic, but it is not the most neutral choice here, and in this sentence it would also need the correct case form.

Hingað means to here / over here / this way, so it shows movement toward the speaker.

This is different from:

• hér = here (location)
• hingað = to here (motion toward here)
• héðan = from here (motion away from here)

So hingað is the right choice because the jacket is being thrown toward the speaker.

Because this is a yes/no question, and in Icelandic those very often begin with the finite verb.

So the structure is:

• Geturðu = Can you
• kastað = throw
• jakkanum mínum = my jacket
• hingað = over here / to here

That verb-first order is completely normal in Icelandic questions.

It can be either, just like English Can you ... ?

In this sentence, it will usually be understood as a request:

• Can you throw my jacket over here?

not a serious question about whether the person is physically capable of doing it.

If you want to sound a bit softer or more polite, Icelandic can also use other patterns, for example:

• Gætirðu kastað jakkanum mínum hingað? = Could you throw my jacket over here?

But Geturðu ... ? is very common and natural.

A rough guide is:

• GEH-tur-thu

A few details:

• stress is on the first syllable
• the ð sound is like the th in this, not thin
• the attached -ðu is a reduced form of þú

So Geturðu is said as one flowing word, not as two strongly separate words.

Kasta is correct and natural for throw.

But in everyday speech, Icelandic speakers also often use henda for throw / toss:

• Geturðu hent jakkanum mínum hingað?

That sounds very natural too.

A useful point: henda, like kasta, also commonly takes the dative, so you still get:

• jakkanum mínum

not a different object form.