Veterinären vill operera klon under återbesöket.

Because vill is a modal verb. In Swedish, modal verbs are normally followed directly by the infinitive, without att.

Examples:

• Jag vill sova = I want to sleep
• Hon kan komma = She can come
• Vi måste gå = We must go

So vill operera is correct.

A useful contrast:

• Veterinären vill operera klon = The veterinarian wants to operate on the claw
• Veterinären vill att någon annan ska operera klon = The veterinarian wants someone else to operate on the claw

In the first sentence, the veterinarian is the one who wants to do the action, so no att is used.

Because veterinären is the definite form: the veterinarian.

Swedish usually puts the definite article at the end of the noun:

• en veterinär = a veterinarian
• veterinären = the veterinarian

So the ending -en is doing the job that English the does.

Klon is the definite singular form of klo.

• en klo = a claw
• klon = the claw

This is a very common Swedish pattern. For many en-words ending in a vowel, the definite singular is formed by adding just -n:

• en sko → skon
• en bro → bron
• en klo → klon

So klon does not mean a different word here; it is simply the claw.

Because Swedish operera can take a direct object in this kind of medical context.

So Swedish says:

• operera knät = operate on the knee
• operera höften = operate on the hip
• operera klon = operate on the claw

English usually needs on, but Swedish often does not. This is a normal difference between the two languages.

Here under means during, not beneath.

So:

• under återbesöket = during the follow-up visit

Swedish under can mean both:

• under bordet = under the table
• under mötet = during the meeting

The context tells you which meaning is intended.

Because återbesök is an ett-word, not an en-word.

• ett återbesök = a follow-up visit / return visit
• återbesöket = the follow-up visit

In Swedish:

• definite singular of many en-words ends in -en
• definite singular of many ett-words ends in -et

Compare:

• en veterinär → veterinären
• ett återbesök → återbesöket

So the ending tells you the grammatical gender of the noun.

It is a compound word:

• åter- = again, back, return
• besök = visit

So återbesök literally means something like return visit or visit again. In medical contexts, it is often best translated as follow-up visit.

This kind of compounding is extremely common in Swedish.

Yes. This is normal Swedish main-clause word order.

The sentence is structured like this:

• Veterinären = subject
• vill = finite verb
• operera = infinitive
• klon = object
• under återbesöket = time expression

A key Swedish rule is V2: in a main clause, the finite verb usually comes in the second position.

That is why this is also possible:

• Under återbesöket vill veterinären operera klon.

Here the time phrase comes first, but vill still stays in second position.

It could look that way to an English speaker, but not in this sentence.

Here klon is clearly from klo and means the claw. The context with veterinären and operera strongly points to an animal’s claw.

So even if the form may remind you of English clone, the intended Swedish word here is klo → klon.

This sentence is a good reminder that å, ä, and ö are separate letters in Swedish, not just decorated versions of a and o.

You see them here in:

• veterinären → ä
• återbesöket → å, ö

Very rough sound hints:

• å often sounds a bit like the vowel in saw
• ä can sound somewhat like the vowel in air or bed, depending on the word and dialect
• ö has no exact English equivalent; German speakers may recognize it more easily

The most important thing for a learner is to treat them as distinct letters with distinct pronunciations and spellings.