Mamy co prawda latarkę czołową, ale nie mamy kurtki przeciwdeszczowej.

This is a very common Polish pattern meaning something like:

• it is true that ..., but ...
• admittedly ..., but ...
• we do have ..., but ...

In your sentence:

• Mamy co prawda latarkę czołową, ale nie mamy kurtki przeciwdeszczowej.

the speaker is conceding one positive fact and then contrasting it with a more important negative fact.

A very natural English equivalent is:

• We do have a headlamp, but we don’t have a rain jacket.

So co prawda introduces a concession, and ale introduces the contrast.

Because Polish often repeats the verb in both parts of a contrast:

• Mamy ... , ale nie mamy ...

Literally: We have ..., but we don’t have ...

This sounds natural and clear in Polish. English sometimes omits the second have, but Polish commonly keeps it.

You might also hear more compressed versions in speech, but the repeated verb is completely standard and often preferred.

Because Polish usually does not need subject pronouns like ja, ty, my, oni unless they are needed for emphasis or contrast.

The verb form already shows the subject:

• mamy = we have

So:

• Mamy already means we have
• Nie mamy already means we do not have

If you said My mamy..., that would sound more emphatic, like we do have...

Because latarkę is the accusative singular form of latarka.

The basic form is:

• latarka = flashlight / torch

After an affirmative verb like mamy (we have), a direct object normally goes into the accusative:

• Mamy latarkę. = We have a flashlight.

So:

• latarka → nominative
• latarkę → accusative

This is very common for feminine nouns ending in -a.

Because Polish usually uses the genitive after a negated verb like nie mamy.

Compare:

• Mamy kurtkę przeciwdeszczową. = We have a rain jacket.
• Nie mamy kurtki przeciwdeszczowej. = We do not have a rain jacket.

So the negation changes the case of the object:

• affirmative mieć
  • accusative
• negative nie mieć
  • genitive

That is why:

• kurtka → kurtkę in the affirmative
• kurtka → kurtki after nie mamy

And the adjective changes with it:

• przeciwdeszczową → przeciwdeszczowej

Because adjectives in Polish must agree with the noun they describe in:

• gender
• number
• case

So:

• latarka czołowa = nominative singular
• latarkę czołową = accusative singular

And:

• kurtka przeciwdeszczowa = nominative singular
• kurtki przeciwdeszczowej = genitive singular

The adjective always follows the noun’s case pattern.

It means headlamp or head torch.

Literally, it comes from:

• latarka = flashlight / torch
• czołowa = frontal / forehead-related / worn on the front of the head

So latarka czołowa is the kind of light you wear on your head.

In natural English, headlamp is the best translation here.

Przeciwdeszczowa means rainproof, for rain, or more naturally in this context, rain as in rain jacket.

It is built from:

• przeciw- = against
• deszcz = rain

So kurtka przeciwdeszczowa is literally something like:

• a jacket against rain

In natural English, that is usually:

• rain jacket
• sometimes waterproof jacket, depending on context

It is best learned as a fixed expression.

Literally:

• co = what
• prawda = truth

But in this phrase, you should not translate word by word. As a set phrase, co prawda means:

• admittedly
• it is true that
• granted

So it is much better to memorize:

• co prawda ... ale ... = admittedly ... but ...

Yes, Polish word order is flexible, and co prawda can appear in slightly different positions, though some sound more natural than others depending on style and emphasis.

For example:

• Mamy co prawda latarkę czołową, ale nie mamy kurtki przeciwdeszczowej.
• Co prawda mamy latarkę czołową, ale nie mamy kurtki przeciwdeszczowej.

Both are natural.

The version in your sentence puts co prawda just after the verb, which is perfectly normal.

Both a and ale can connect contrasting ideas, but ale is stronger and corresponds more directly to but.

Here the speaker is making a clear contrast:

• we do have one useful thing
• but we do not have another important thing

So ale fits very well.

Very roughly:

• a = and/but, often a milder contrast
• ale = but, stronger and clearer

Yes. In Polish, nie is written separately from finite verb forms.

So:

• mamy = we have
• nie mamy = we do not have

This is normal for verbs in Polish.

Sometimes, yes, especially in casual speech, but the full version is clearer and more standard for learners:

• Mamy co prawda latarkę czołową, ale nie mamy kurtki przeciwdeszczowej.

A native speaker might sometimes shorten things in conversation if the context is obvious, but this complete version is natural, grammatical, and easy to understand.

So for learning purposes, this sentence is a very good model.