Po pracy tata otworzył słoik z miodem i powiedział, że chętnie zje jeszcze jedną kromkę chleba.

Because po in the sense of after normally requires the locative case.

• praca = work
• locative singular: pracy

So:

• po pracy = after work

This is a very common pattern in Polish:

• po obiedzie = after dinner
• po szkole = after school
• po filmie = after the film

Great question. Tata is one of those Polish nouns that looks feminine because it ends in -a, but it refers to a male person, so it takes masculine agreement.

That is why you get:

• tata otworzył = dad opened
• tata powiedział = dad said
• miły tata = nice dad

Not:

• tata otworzyła
• tata powiedziała

The same happens with some other male-person words like mężczyzna.

Otworzył is the past tense, masculine singular form of otworzyć (to open).

It matches tata:

• masculine singular subject → otworzył

Polish past tense shows gender in the singular:

• otworzył = he opened
• otworzyła = she opened
• otworzyło = it opened

So tata otworzył literally means dad opened.

Here z means with, and in that meaning it takes the instrumental case.

• miód = honey
• instrumental singular: miodem

So:

• słoik z miodem = a jar with honey

This describes the jar as containing honey.

Be careful: z can also mean from, but then it often takes the genitive. Here it clearly means with.

Yes, and both are natural, but they are not exactly identical in feel.

• słoik z miodem = a jar with honey
• słoik miodu = a jar of honey

The version in your sentence uses z + instrumental, which emphasizes the jar together with its contents.
The genitive version (słoik miodu) is also very common and often sounds a bit more like English a jar of honey.

Both are good Polish.

In Polish, a clause introduced by że (that) is normally separated by a comma.

So:

• powiedział, że... = he said that...

This is standard Polish punctuation. English often omits that, but in Polish że is very common, and the comma is expected.

Że introduces a subordinate clause and usually means that.

So this part:

• powiedział, że chętnie zje...

means:

• he said that he would gladly eat...

It connects the reporting verb (powiedział) with the content of what was said.

Chętnie means something like:

• gladly
• willingly
• with pleasure

So:

• chętnie zje = he will gladly eat / he’d happily eat

It often expresses willingness or readiness:

• Chętnie pomogę. = I’ll gladly help.
• Chętnie to zrobię. = I’ll be happy to do it.

It does not mean exactly hungrily or eagerly, though in context it can sound enthusiastic.

Because zje comes from the perfective verb zjeść, while je comes from the imperfective verb jeść.

In Polish:

• jeść = to eat, ongoing/repeated process
• zjeść = to eat up / to eat a whole portion, completed action

The form zje is formally a present-tense form, but for a perfective verb it usually refers to the future:

• zje = he will eat / he will eat up

So chętnie zje jeszcze jedną kromkę chleba suggests a completed portion: one more slice.

Morphologically, it looks like a present-tense form, but functionally it expresses the future, because it belongs to a perfective verb.

This is a key Polish pattern:

• imperfective je = he is eating / he eats
• perfective zje = he will eat (up)

So in practice, you should understand zje here as future.

Because they are the direct object of zje, so they are in the accusative case.

Base forms:

• jedna = one
• kromka = slice

Accusative singular feminine:

• jedną
• kromkę

So:

• zje jedną kromkę = he will eat one slice

Since kromka is feminine, the numeral/adjective must agree with it.

Because chleba is in the genitive case after a noun of portion or measure.

• kromka = slice
• chleb = bread
• genitive singular: chleba

So:

• kromka chleba = a slice of bread

This is very common in Polish:

• szklanka wody = a glass of water
• filiżanka herbaty = a cup of tea
• kawałek sera = a piece of cheese

The second noun often goes into the genitive to mean of ...

Here jeszcze means another / one more.

So:

• jeszcze jedną kromkę chleba = one more slice of bread

But jeszcze is a flexible word and can also mean still or yet, depending on context:

• Jeszcze śpi. = He’s still sleeping.
• Jeszcze nie wiem. = I don’t know yet.
• Jeszcze jedną kawę? = One more coffee?

In your sentence, because it comes before jedną kromkę, the meaning is clearly one more.

Kromka specifically means a slice of bread. It is the natural word when talking about bread cut into slices.

• kromka chleba = slice of bread

Kawałek is broader:

• kawałek chleba = a piece of bread

So kromka is more precise here and sounds very natural.

Polish word order is fairly flexible, though the original sentence is completely natural.

The given order:

• Po pracy tata otworzył słoik z miodem i powiedział, że chętnie zje jeszcze jedną kromkę chleba.

is neutral and smooth.

Possible variations are also grammatical, for example:

• Tata po pracy otworzył słoik z miodem...
• Po pracy tata powiedział, że chętnie zje jeszcze jedną kromkę chleba, i otworzył słoik z miodem.

But changing the order can slightly change the emphasis or flow. Polish often uses word order for information structure rather than strict grammatical necessity.

Because the sentence reports what he said about a future action.

He spoke in the past:

• powiedział = he said

But what he said concerns what he will do next:

• zje = he will eat

This works just like English:

• He said that he would eat another slice of bread.

Polish often keeps the future meaning directly in the subordinate clause.

A fairly literal breakdown would be:

• Po pracy = after work
• tata = dad
• otworzył = opened
• słoik z miodem = a jar with honey
• i powiedział = and said
• że = that
• chętnie = gladly
• zje = will eat
• jeszcze jedną = one more
• kromkę chleba = slice of bread

So a literal version would be something like:

After work dad opened a jar with honey and said that he would gladly eat one more slice of bread.

Yes, a few common ones:

• słoik: the ł is pronounced like English w, so roughly swoyik
• miodem: io is pronounced as two sounds, roughly myo-dem
• chętnie: ch is like a harsh h sound, and ę before t is often pronounced approximately like en, so roughly hen-tnie with a Polish sound, not an English h
• jedną: the ą at the end is nasal; before pauses it often sounds somewhat like om/on, but not exactly
• kromkę: the final ę is also nasalized, often weakened in everyday speech

A big general tip: don’t try to pronounce Polish nasal vowels exactly like French nasal vowels. Their pronunciation changes depending on the following sound.