אני צריכה עותק נוסף של המסמך.

Because the speaker is female. In this kind of sentence, צריך / צריכה agrees with the person who needs something.

• אני צריך = I need (said by a male)
• אני צריכה = I need (said by a female)

So if a man were saying the same sentence, it would be אני צריך עותק נוסף של המסמך.

Usually אני is needed, because צריכה only shows gender and number, not person.

By itself, צריכה could mean:

• I need (female)
• you need (female, singular)
• she needs

So אני makes it clear that the meaning is I need.

Hebrew normally leaves out the verb to be in the present tense.

So where English says I am hungry, I am tired, or I need, Hebrew often just says:

• אני רעבה = I am hungry
• אני עייפה = I am tired
• אני צריכה = I need

That omission is completely normal in Hebrew.

Because in Hebrew, adjectives usually come after the noun they describe.

So:

• עותק נוסף = an additional copy
• literally: copy additional

Also, the adjective has to agree with the noun in gender, number, and definiteness.

Here:

• עותק is masculine singular
• so the adjective is נוסף, also masculine singular

If the noun were feminine, the adjective would change too.

Because עותק is a masculine noun.

So the adjective has to match it:

• עותק נוסף = masculine singular
• if the noun were feminine, you would use נוספת

For example:

• גרסה נוספת = an additional version

נוסף means additional, extra, or another depending on context.

In this sentence, עותק נוסף is best understood as:

• an additional copy
• another copy
• an extra copy

A useful nuance:

• נוסף = additional / extra
• אחר = another / different
• עוד = more / another / one more

So עותק נוסף sounds very natural for an additional copy, especially in office or document-related contexts.

Because של is the common Hebrew way to say of or belonging to.

So:

• עותק של המסמך = a copy of the document

In the full sentence:

• עותק נוסף של המסמך = an additional copy of the document

This is a very normal Hebrew structure.

Because ה־ means the.

• המסמך = the document
• מסמך = a document / document

Hebrew has a definite article (ה־), but it does not have an indefinite article like a/an.

So:

• עותק can mean a copy
• העותק means the copy

That is why the sentence has עותק but המסמך.

A common pronunciation is:

ani tzrikhá ótek nosáf shel hamismách

A few notes:

• אני = ani
• צריכה = tzrikhá
• עותק = ótek
• נוסף = nosáf
• של = shel
• המסמך = hamismách

The כ/ך sound in צריכה and המסמך is the throat sound like ch in Scottish loch or German Bach.

Yes. That is also very natural.

• אני צריכה עותק נוסף של המסמך = I need an additional copy of the document
• אני צריכה עוד עותק של המסמך = I need another copy of the document

Both are good.
עוד עותק is often a little more everyday and conversational, while עותק נוסף can sound slightly more formal or precise.