Moja kuzynka jest fryzjerką i pracuje blisko rynku.

Because after jest in this kind of sentence, Polish usually uses the instrumental case for professions, roles, and identities.

• fryzjerka = hairdresser (dictionary form, nominative)
• jest fryzjerką = is a hairdresser

So:

• Moja kuzynka jest fryzjerką = My cousin is a hairdresser

This is a very common pattern in Polish:

• On jest nauczycielem = He is a teacher
• Ona jest lekarką = She is a doctor

It is in the instrumental singular.

The base form is fryzjerka.
In the instrumental singular, many feminine nouns ending in -a change to -ą:

• fryzjerka → fryzjerką
• nauczycielka → nauczycielką
• koleżanka → koleżanką

In this sentence, it is instrumental because it follows jest in the meaning is a hairdresser.

Because blisko normally requires the genitive case.

• rynek = market square / market (base form)
• rynku = genitive singular

So:

• blisko rynku = near the market square

This same thing happens with some other prepositions and expressions too:

• blisko domu = near the house
• blisko szkoły = near the school
• blisko centrum = near the center

So the pattern is:

• blisko + genitive

Here rynek usually means the market square or town square, especially in a Polish town or city context.

So blisko rynku usually means:

• near the town square
• near the main square

It does not necessarily mean a modern supermarket or a general economic market. In everyday Polish, rynek often refers to the central square in a town.

Moja means my.

It has this form because it agrees with kuzynka, which is:

• feminine
• singular
• nominative

So:

• mój = my (masculine)
• moja = my (feminine)
• moje = my (neuter)

Examples:

• mój brat = my brother
• moja siostra = my sister
• moje dziecko = my child

In Moja kuzynka, both words match grammatically.

Kuzynka means female cousin.

The masculine form is:

• kuzyn = male cousin

So:

• moja kuzynka = my female cousin
• mój kuzyn = my male cousin

English uses cousin for both, but Polish distinguishes gender here.

Because Polish does not have articles like a/an/the.

So where English says:

• She is a hairdresser

Polish simply says:

• Ona jest fryzjerką

There is no separate word for a.

This is normal in Polish. The meaning of a, an, or the is understood from context.

Because Polish often omits subject pronouns when they are already clear from the verb form or context.

• pracuje already tells you it is he/she/it works
• the noun moja kuzynka already tells you who the subject is

So Polish does not need to say ona here.

You could say:

• Moja kuzynka jest fryzjerką i ona pracuje blisko rynku

but it sounds more emphatic than necessary. The natural version is the original one:

• Moja kuzynka jest fryzjerką i pracuje blisko rynku

Pracuje is the 3rd person singular present tense of pracować = to work.

So:

• ja pracuję = I work
• ty pracujesz = you work
• on/ona/ono pracuje = he/she/it works

In the sentence, it agrees with moja kuzynka, so it means:

• she works

Depending on context, Polish present tense can translate as either:

• works
• is working

Here works is the most natural translation.

The given word order is natural, but Polish word order is more flexible than English.

Original:

• Moja kuzynka jest fryzjerką i pracuje blisko rynku.

You could also say:

• Moja kuzynka pracuje blisko rynku i jest fryzjerką.

That is grammatical, but it changes the focus a little. The original version first gives her profession, then adds where she works.

Polish word order often changes for emphasis, style, or information structure, but the original sentence is a very normal neutral order.

Yes, in theory it can, because blisko means near/close to and rynek can have more than one meaning.

But in everyday Polish, especially in a sentence about where someone works, rynek very often suggests the town square.

So a learner should usually understand:

• blisko rynku = near the town square / market square

If context were different, it could sometimes mean something else, but this is the most likely meaning.

Then several words would change to match masculine gender:

• Moja kuzynka jest fryzjerką i pracuje blisko rynku.
• Mój kuzyn jest fryzjerem i pracuje blisko rynku.

Changes:

• moja → mój
• kuzynka → kuzyn
• fryzjerką → fryzjerem

This shows how important gender agreement is in Polish.