Pani Anna mówi coraz lepiej po polsku.

Pani means Ms./Mrs. and is a polite way to address or refer to a woman. So Pani Anna is a respectful, semi-formal way to say Ms. Anna or Anna politely.

In Polish, using Pan/Pani + first name is very common in polite conversation, more common than in English.
So:

• Anna = just Anna
• Pani Anna = polite/respectful Anna
• Pani alone can also mean you when speaking formally to a woman

In this sentence, Pani Anna is simply the subject.

Mówi is the 3rd person singular present tense form of the verb mówić (to speak / to say).

So:

• ja mówię = I speak / am speaking
• ty mówisz = you speak
• on/ona/Pani mówi = he/she/you(formal) speaks

Because the subject is Pani Anna (she), Polish uses mówi.

It can mean both, depending on context.

Polish present tense often covers both:

• she speaks
• she is speaking

In this sentence, because of coraz lepiej (better and better / more and more well), the natural meaning is something like:

• Ms. Anna speaks Polish better and better
• Anna is speaking Polish more and more well

In English, speaks better and better sounds most natural.

Coraz lepiej means better and better or more and more well.

It is a very common Polish pattern:

• coraz + comparative

Examples:

• coraz lepiej = better and better
• coraz szybciej = faster and faster
• coraz więcej = more and more
• coraz mniej = less and less

So Pani Anna mówi coraz lepiej po polsku means that her Polish is improving over time.

Because lepiej is the comparative form of the adverb dobrze (well).

This is an irregular pattern:

• dobrze = well
• lepiej = better
• najlepiej = best

Here we need an adverb, not an adjective, because it describes how she speaks.

Compare:

• dobry nauczyciel = a good teacher
  
  • adjective describing a noun
• mówi dobrze = speaks well
  
  • adverb describing a verb

So lepiej is correct because it modifies mówi.

In Polish, one common way to say that someone speaks a language is:

• mówić po + language form

So:

• mówić po polsku = to speak Polish
• mówić po angielsku = to speak English
• mówić po niemiecku = to speak German

This structure is different from English. You do not usually say the language name here as a direct object the way English does.

So Polish uses:

• po polsku

rather than something like polski.

After the preposition po in this meaning, Polish uses the locative case.

That is why:

• polski becomes polsku

Similarly:

• angielski → po angielsku
• niemiecki → po niemiecku
• francuski → po francusku

So po polsku literally follows the grammar pattern required by po.

Because in Polish, names of languages are normally written with a lowercase letter.

So:

• polski
• angielski
• niemiecki

are lowercase.

This is different from English, where Polish, English, and German are capitalized.

So po polsku is written correctly with lowercase p.

Yes. Polish word order is more flexible than English word order.

The neutral order here is:

• Pani Anna mówi coraz lepiej po polsku.

But you could also hear variations like:

• Pani Anna coraz lepiej mówi po polsku.
• Po polsku Pani Anna mówi coraz lepiej.

These versions may shift emphasis slightly, but the basic meaning stays the same.

The original sentence sounds natural and neutral.

It is singular.

Even though Pani can function as a formal you, when referring to one woman it still takes 3rd person singular verb forms:

• Pani mówi = you (formal, to one woman) speak

So with Pani Anna, we use:

• mówi, not mówią

Yes, depending on context.

It can mean:

• knowing/using the Polish language in general
• speaking in Polish in a particular situation

For example:

• Ona mówi po polsku. = She speaks Polish.
• Proszę mówić po polsku. = Please speak in Polish.

In your sentence, the meaning is clearly about ability/improvement in the language.

Because Polish often leaves subject pronouns out when they are not needed.

The verb form plus the noun subject already tells us who is doing the action:

• Pani Anna mówi = Ms. Anna speaks

Adding ona would usually be unnecessary unless you want contrast or emphasis.

Polish commonly avoids pronouns when the subject is already clear.