Potem muszę wypełnić także drugi wniosek dla mojej siostry.

Muszę is the 1st person singular form of the verb musieć (to have to / must).

So:

• muszę = I have to / I must
• musisz = you have to
• musi = he/she/it has to

In this sentence, muszę matches the implied subject I.

This is about aspect, which is very important in Polish.

• wypełniać = imperfective: the process of filling out, doing it in general, repeatedly, or without focusing on completion
• wypełnić = perfective: to fill out completely, with the result in mind

After muszę, Polish can use either aspect depending on meaning:

• Muszę wypełniać formularze codziennie. = I have to fill out forms every day.
• Muszę wypełnić ten formularz. = I have to fill out this form (and finish it).

In your sentence, the idea is completing a specific application, so wypełnić is natural.

Także means also / too / as well.

It is very close in meaning to też, and in many sentences they can both work:

• Muszę także wypełnić drugi wniosek.
• Muszę też wypełnić drugi wniosek.

Both mean roughly the same thing.

A small difference:

• też is usually a bit more common and conversational
• także can sound a little more formal or written

So here, także simply adds the idea in addition.

It is the object, so it is in the accusative case. But for many masculine inanimate nouns, the accusative singular looks exactly like the nominative singular.

So:

• nominative: drugi wniosek
• accusative: drugi wniosek

That is why nothing seems to change.

If this were a masculine animate noun, you would see a clearer difference.

Because the preposition dla requires the genitive case.

So:

• moja siostra = nominative (my sister)
• dla mojej siostry = genitive (for my sister)

Both words change because adjectives, possessives, and nouns must agree in case:

• moja → mojej
• siostra → siostry

This is a very common pattern in Polish: the preposition controls the case.

Here dla means for.

So dla mojej siostry means for my sister.

It is used to show who something is intended for or who benefits from it. In this sentence, the application is being filled out for the sister.

Just remember:

• dla + genitive

Examples:

• prezent dla mamy = a present for mom
• list dla kolegi = a letter for a friend
• wniosek dla mojej siostry = an application for my sister

It can suggest either, depending on context.

Most literally, drugi means second. But in natural English translation, it may sometimes sound better as another:

• drugi wniosek = a second application
• sometimes naturally: another application

The Polish word still implies that there is one application already, and this is the next one.

Yes, Polish word order is fairly flexible, and changing it changes the focus or emphasis, not usually the basic meaning.

The given sentence is natural:

• Potem muszę wypełnić także drugi wniosek dla mojej siostry.

Possible alternatives:

• Muszę potem wypełnić także drugi wniosek dla mojej siostry.
• Także drugi wniosek dla mojej siostry muszę potem wypełnić.
  This is more marked and emphasizes the second application.

Why is Potem first?
Because it sets the time frame right away: Then / After that...

So the original order is neutral and natural.

They are similar, but not always identical.

• potem = then / after that / later
• później = later

In many contexts they can overlap:

• Potem muszę...
• Później muszę...

But potem often connects more naturally to a sequence of events:
first this, then that

So in your sentence, Potem sounds very natural if the speaker is listing steps.

Yes. Wypełnić wniosek is a very common and natural collocation.

It means to fill out / complete an application.

Other similar expressions:

• wypełnić formularz = fill out a form
• złożyć wniosek = submit an application
• napisać wniosek = write an application

So be careful:
wypełnić means completing the form itself, not handing it in.

Yes, very possibly.

Dla mojej siostry literally means for my sister, but in context it can imply different things, such as:

• the application concerns the sister
• the speaker is filling it out on her behalf
• the application is intended for her benefit

The exact nuance depends on the situation, but all of those are compatible with the Polish sentence.

Wniosek is pronounced approximately like:

vnyo-sek

A few tips:

• w in Polish sounds like English v
• wni is pronounced together, not with a separate vowel before it
• io here gives a yo-like sound

So the word begins with something close to vnyo-.

This kind of consonant cluster is normal in Polish, even though it may feel awkward at first to English speakers.

Yes. In most cases, muszę can be translated as either:

• I must
• I have to

In everyday English, I have to is often more natural.
In Polish, muszę covers both ideas very well.

So:

• Muszę wypełnić drugi wniosek.
• I have to fill out a second application.
• I must fill out a second application.

All are fine, with only a slight stylistic difference in English.