Hvolpurinn gengur í taumi eftir að hafa klárað hundanámskeiðið.

The base noun is hvolpur, meaning puppy. Icelandic usually puts the definite article at the end of the noun instead of using a separate word like English the.

So:

• hvolpur = puppy
• hvolpurinn = the puppy

In this sentence, hvolpurinn is also in the nominative singular, because it is the subject of the sentence.

Ganga is the dictionary form, meaning to walk.
Gengur is the 3rd person singular present tense form, used because the subject is singular: hvolpurinn.

So:

• að ganga = to walk
• ég geng = I walk
• hann/hún/það gengur = he/she/it walks

This verb is irregular enough that the stem changes:

• ganga → gengur

That kind of vowel change is very common in Icelandic strong verbs.

Í taumi means on a leash or on a lead.

Literally, it looks more like in a leash/lead, but the natural English translation is on a leash. This is a standard Icelandic expression for a dog being led with a leash.

The noun is:

• taumur = rein, lead, leash

So ganga í taumi is a natural phrase meaning to walk on a leash.

Because taumi is the dative singular form of taumur.

The preposition í can take two different cases:

• accusative for motion into
• dative for location or state

Here the puppy is not moving into a leash; it is walking while being on a leash. So Icelandic uses the dative:

• í taumi

This is also just the normal fixed form of the expression ganga í taumi.

It means after having finished.

Breakdown:

• eftir að = after
• hafa = to have
• klárað = finished/completed

So:

• eftir að hafa klárað hundanámskeiðið = after having finished the dog-training course

This is a very common Icelandic structure when the subject of the after-phrase is understood to be the same as the subject of the main clause.

Here klárað is the form used with hafa in a perfect-style construction. It is often called the supine in Icelandic grammar.

The basic verb is:

• að klára = to finish, to complete

In this sentence:

• hafa klárað = to have finished

For many weak verbs, this form looks the same as the neuter singular past participle, so learners often notice it as a past-participle-looking form.

Also, klára is a very common everyday verb. In more formal language, Icelandic may sometimes use other verbs such as ljúka, depending on context.

Because Icelandic often builds compounds using a special linking form for the first noun. Here the word is:

• hundanámskeiðið

This breaks down as:

• hunda- = a compound form related to hundur (dog)
• námskeið = course
• -ið = the

So the whole word means the dog-training course or the dog course.

For English speakers, the important thing is that Icelandic very often makes this kind of idea into one compound word, not several separate words.

Because the whole compound functions as one noun.

In Icelandic, when a noun is a compound, the suffixed definite article normally goes on the entire compound, at the end:

• námskeið = course
• námskeiðið = the course
• hundanámskeið = dog-training course
• hundanámskeiðið = the dog-training course

So Icelandic does not usually mark each part separately. The compound is treated as one unit.

Normally, it is understood to be the same subject as in the main clause.

So in this sentence, the default reading is that the puppy is the one that completed the course.

That is why the infinitive construction works smoothly here:

• Hvolpurinn gengur í taumi eftir að hafa klárað hundanámskeiðið.

If you wanted a different subject, Icelandic would usually make that explicit with a full clause, for example:

• eftir að eigandinn hafði klárað hundanámskeiðið
  = after the owner had finished the dog-training course

It can potentially cover both, because Icelandic present tense often overlaps with both the English simple present and present progressive.

So gengur can mean:

• walks
• is walking

In this sentence, the most natural meaning is probably more general or habitual:

• The puppy walks on a leash
• or even The puppy now knows how to walk on a leash

That fits well with the phrase after having finished the dog-training course. The sentence sounds less like a one-time action and more like a learned behavior or result.