Proszę mówić trochę głośniej podczas lekcji.

This is a very common Polish way to make a polite request or instruction:

• proszę + infinitive

So proszę mówić means something like please speak or please do speak.

It is often used:

• when speaking politely to one person
• when addressing a group
• in formal situations
• in classroom or service contexts

Compare:

• Mów głośniej. = Speak louder. (informal, to one person)
• Mówcie głośniej. = Speak louder. (informal, to more than one person)
• Proszę mówić głośniej. = Please speak louder. (polite/formal)

Also, proszę is literally the 1st person singular form of prosić (to ask/request), but in sentences like this it functions like please.

Because mówić is imperfective, and here Polish wants the idea of speaking in an ongoing way, not saying one thing once.

• mówić = to speak, to be speaking, to speak habitually/continuously
• powiedzieć = to say, to tell, to say once and complete the action

In this sentence, the speaker is asking someone to speak louder during the lesson, so it is about the manner of speaking over a period of time. That is why mówić is the natural choice.

If you used powiedzieć, it would sound more like asking someone to say something once.

Głośniej is the comparative adverb of głośno.

• głośno = loudly
• głośniej = more loudly / louder

Since it modifies the verb mówić (to speak), Polish uses an adverb, not an adjective.

Compare:

• Mówić głośno = to speak loudly
• Mówić głośniej = to speak louder

A useful contrast:

• głośny / głośniejszy = adjective, used with nouns
  
  • głośniejszy głos = a louder voice
• głośno / głośniej = adverb, used with verbs
  
  • mówić głośniej = to speak louder

Trochę means a bit or a little.

Here it softens the request:

• Proszę mówić głośniej. = Please speak louder.
• Proszę mówić trochę głośniej. = Please speak a little louder.

So it makes the sentence sound:

• less blunt
• more natural
• more polite

English often does the same thing with a bit, a little, or slightly.

Because the preposition podczas requires the genitive case.

The base form is:

• lekcja = lesson

But after podczas, it changes to:

• podczas lekcji = during the lesson

So this is a case issue, not a special vocabulary item.

A few similar examples:

• podczas obiadu = during dinner
• podczas spotkania = during the meeting
• podczas zajęć = during classes

Both can often be translated as during class / in class, but they are not exactly the same in feel.

• podczas lekcji = during the lesson
  
  • slightly more explicit
  • a bit more formal or careful
• na lekcji = in class / during class
  
  • very common in everyday speech
  • often a little more conversational

So:

• Proszę mówić trochę głośniej podczas lekcji.
• Proszę mówić trochę głośniej na lekcji.

Both are understandable and natural, but podczas lekcji sounds a bit more formal and precise.

It can be either, depending on context.

That is one useful thing about proszę + infinitive: it does not mark number the way imperative forms do.

So Proszę mówić... may be said:

• politely to one person
• politely to several people
• in a general instructional way

If Polish wants to make it clearer, it can add:

• Proszę pana mówić... (to a man)
• Proszę panią mówić... (to a woman)
• Proszę państwa mówić... (to a group, though this exact wording is less common in many contexts)

But usually the context is enough.

Yes, but the tone would change.

Examples:

• Mów trochę głośniej podczas lekcji.
  = Speak a little louder during the lesson.
  (informal, to one person)

• Mówcie trochę głośniej podczas lekcji.
  = Speak a little louder during the lesson.
  (informal, to more than one person)

• Proszę mówić trochę głośniej podczas lekcji.
  = Please speak a little louder during the lesson.
  (polite/formal)

So the version with proszę is softer and more polite.

Not completely. Polish word order is fairly flexible, and this sentence could be rearranged.

For example:

• Proszę mówić trochę głośniej podczas lekcji.
• Podczas lekcji proszę mówić trochę głośniej.
• Proszę podczas lekcji mówić trochę głośniej.

These all make sense.

The original order is very natural. It gives the main request first:

• please speak a little louder and then adds the time/context:
• during the lesson

So the word order is chosen for natural flow, not because it is the only grammatical option.

Because Polish very often omits subject pronouns when they are not needed.

English usually needs you:

• Please speak a little louder during the lesson.

Polish does not need an explicit ty / wy / pan / pani / państwo here, because the request structure already makes it clear that someone is being addressed.

In fact, adding a pronoun is often unnecessary unless you want:

• contrast
• emphasis
• extra clarity

So leaving out you is completely normal Polish.