Czy masz może jakiś wolny karton? Przyda mi się przy przeprowadzce.

Czy turns the sentence into a yes/no question.

• Masz jakiś wolny karton. = You have some spare box. / You have a spare box.
• Czy masz jakiś wolny karton? = Do you have a spare box?

In everyday Polish, czy is often optional, especially in speech. You could also just say:

• Masz może jakiś wolny karton?

That is still a perfectly natural question.

Here może does not literally mean maybe in the usual sense. It is often used to soften a request or question.

So:

• Masz jakiś wolny karton? = direct
• Masz może jakiś wolny karton? = softer, more polite, more tentative

A natural English equivalent might be:

• Do you happen to have a spare box?
• Any chance you have a spare box?

So może here adds a friendly, non-pushy tone.

Jakiś means something like some, any, or a certain depending on context.

In this sentence:

• jakiś wolny karton = some spare box / any spare box

The speaker is not asking for a specific box, just any one that would work.

This is very common in Polish when asking for something indefinite:

• Masz jakiś długopis? = Do you have a pen / any pen?
• Znasz jakąś dobrą kawiarnię? = Do you know a good café / any good café?

Because karton is a masculine inanimate noun, and in the singular its accusative form is the same as the nominative.

So:

• nominative: karton
• accusative: karton

Since masz takes a direct object, we need the accusative, but for this noun it looks unchanged.

Compare:

• Mam karton. = I have a box.
• Widzę karton. = I see a box.

But for masculine animate nouns, accusative would change:

• mam kota = I have a cat

Here wolny means free, available, or spare, not free of charge.

So wolny karton means:

• a spare box
• an unused box
• an available box

In English, free box could be misunderstood as a box that costs nothing, but in Polish wolny often means not occupied / available / not being used.

For example:

• Mam wolny wieczór. = I have a free evening.
• Czy to miejsce jest wolne? = Is this seat free?

So here the meaning is Do you have any spare/available cardboard box?

Both can be translated as box, but they are not exactly the same.

• karton usually means a cardboard box
• pudełko is a more general box, often smaller, like a gift box, shoe box, or container

Because the second sentence mentions moving, karton is especially natural, since cardboard moving boxes are what people usually mean.

So:

• karton do przeprowadzki = moving box / cardboard box for moving

Przydać się is a very common Polish verb meaning:

• to come in handy
• to be useful
• to be useful to someone

So:

• Przyda mi się. = It will come in handy for me. / It’ll be useful to me.

This expression sounds very natural in Polish and is often preferred when you want to say something would be useful, rather than strongly stating that you need it.

Compare:

• Będę potrzebować kartonu. = I will need a box.
• Karton przyda mi się. = A box would come in handy / be useful to me.

The second version sounds a bit softer and less demanding.

They do different jobs.

• mi = to me / for me
• się = part of the verb przydać się

So the whole structure is:

• coś przyda się komuś = something will be useful to someone

Examples:

• Ten kabel przyda mi się. = This cable will come in handy for me.
• Ta rada przyda ci się. = This advice will be useful to you.
• To może przydać się wszystkim. = This may be useful to everyone.

Important: się here is not really something you can just remove. The verb is normally learned as przydać się.

Because przydać się is a perfective verb, and perfective verbs in Polish do not have a normal present tense meaning. Their present-looking forms usually refer to the future.

So:

• przyda mi się literally looks present
• but it means it will be useful / it will come in handy

This is very common with perfective verbs:

• zadzwonię = I will call
• zrobię = I will do
• przyda się = it will come in handy

So in this sentence, Przyda mi się naturally means I’ll be able to use it / It’ll come in handy for me.

Przy przeprowadzce means something like:

• during the move
• for the move
• when moving
• in connection with the move

The noun is:

• przeprowadzka = a move / moving house

After przy, Polish uses the locative case, so:

• przeprowadzka → przeprowadzce

This use of przy does not only mean next to or by. It can also mean in the context of or during.

Examples:

• przy pracy = at work / while working
• przy obiedzie = during lunch
• przy przeprowadzce = during a move / for moving

Yes. It is grammatically a question, but pragmatically it functions as a polite request.

When someone says:

• Czy masz może jakiś wolny karton?

they are not usually just gathering information. They are often indirectly asking:

• Could you give me one if you have one?

This is very common in both Polish and English. Asking whether someone has something can be a softer way of asking for it.

Yes, it sounds very natural and polite.

Why it sounds polite:

• czy makes it a question
• może softens the tone
• jakiś makes it less demanding
• Przyda mi się przy przeprowadzce explains the reason, which also makes the request sound more considerate

So the overall effect is friendly and normal, like:

• Do you happen to have a spare box? It would come in handy for my move.

Yes. Polish word order is flexible, though different orders can sound slightly different in emphasis.

For example:

• Czy masz może jakiś wolny karton?
• Masz może jakiś wolny karton?
• Jakiś wolny karton masz może? — possible, but more marked

The most neutral and natural version is the one you were given.

In the second sentence, you could also say:

• Przy przeprowadzce przyda mi się.
• Przyda mi się przy przeprowadzce.

The original order is the most natural, with the main message first and the reason after it.