Reggel sapkát és pulóvert veszek fel, mert hűvös a levegő.

The -t is the accusative ending, used for the direct object of the verb.

In this sentence, the things being put on are:

• sapkát = a hat / hat
• pulóvert = a sweater / sweater

The base forms are:

• sapka → sapkát
• pulóver → pulóvert

Because they are the objects of veszek fel (I put on), they take the accusative.

Because felvesz is a verb with a prefix: fel- + vesz.

Here, felvesz means to put on clothing.
Without the prefix, vesz usually means to buy or to take.

So:

• veszek = I buy / I take
• felveszek or veszek fel = I put on

In this sentence, the prefix is separated:

• pulóvert veszek fel

That is completely normal in Hungarian.

Yes, both are possible, but the word order changes with emphasis and sentence structure.

Hungarian verbal prefixes often:

• stay attached: felveszek
• or get separated: veszek fel

The separated form often appears when something else is in focus or when the sentence flows that way.

So these are both possible:

• Reggel sapkát és pulóvert felveszek...
• Reggel sapkát és pulóvert veszek fel...

The version in your sentence sounds natural and idiomatic.

This is a very common question.

• felveszek / veszek fel = I put on
• viselek = I wear / I am wearing

So:

• Reggel sapkát és pulóvert veszek fel = In the morning I put on a hat and a sweater
• Sapkát és pulóvert viselek = I am wearing a hat and a sweater

The first focuses on the action of dressing.
The second focuses on what you have on.

Hungarian usually does not need subject pronouns unless you want emphasis or contrast.

The verb ending already shows the subject:

• veszek = I take / I buy / I put on
• the -ek ending tells you it is first person singular (I)

So:

• veszek already means I put on
• én veszek would only be used for emphasis, like I am the one who puts it on

Reggel here means in the morning or in the morning time.

It is often used without a preposition in English and without an article in Hungarian:

• Reggel = in the morning
• Este = in the evening
• Éjjel = at night

So Hungarian often uses a simple time word where English uses a prepositional phrase.

Hungarian often uses the singular when talking about individual items of clothing being worn or put on.

Here it means one hat and one sweater:

• sapkát = a hat
• pulóvert = a sweater

If you wanted plural, you would use plural forms, but that would mean multiple hats or multiple sweaters, which does not fit the normal meaning here.

Both are possible, but they are not exactly the same in emphasis.

• Hűvös a levegő puts emphasis on hűvös (cool/chilly). It sounds like The air is cool with focus on the quality.
• A levegő hűvös is more neutral, more like simply stating The air is cool.

Hungarian word order is flexible and often shows focus or what is being emphasized.

In this sentence, hűvös a levegő sounds very natural.

Because levegő means air, and Hungarian often uses the definite article when referring to the air in the general surrounding sense.

So:

• a levegő = the air

This is natural in Hungarian, even when English might also say simply the air in the same context:

• hűvös a levegő = the air is cool

Hűvös usually means cool, chilly, or slightly cold, not strongly cold.

So it is softer than:

• hideg = cold

Compare:

• Hűvös a levegő = The air is cool / chilly
• Hideg a levegő = The air is cold

Here hűvös suggests weather cool enough that you want a hat and sweater, but not necessarily freezing.

Veszek is:

• present tense
• first person singular
• from the verb vesz

So it means I take / I buy, and with fel it means I put on.

In this sentence:

• veszek fel = I put on

Hungarian often leaves out the article when English would use a.

So:

• sapkát és pulóvert veszek fel literally looks like
• I put on hat and sweater

But natural English translation is:

• I put on a hat and a sweater

This is normal Hungarian usage, especially when introducing nonspecific objects.

It can often sound habitual or like a general statement, especially because of Reggel (in the morning).

So it may mean:

• In the morning I put on a hat and a sweater or
• In the mornings I put on a hat and a sweater

Hungarian present tense often covers both simple present and habitual meanings, depending on context.

Yes, in basic coordination és works like English and.

Here:

• sapkát és pulóvert = a hat and a sweater

It simply joins the two objects of the verb.