Learn Hardcore Hausa: A kasuwa akwai mutane masu yawa. - At the market there are many people.

The first version puts the location a kasuwa (“at the market”) at the beginning for emphasis or to set the scene: “As for the market, there are many people (there).”
The second is closer to English “There are many people at the market” and is also very common.
So the sentence begins with A kasuwa simply to highlight the location first, not because of a strict rule.

What does a mean in A kasuwa?

A is a preposition that usually means “in / at / on”, depending on context.

a kasuwa – at/in the market
a gida – at home
a makaranta – at school

It does not change for gender or number.
English often distinguishes “in” vs. “at”, but Hausa a covers both, and context tells you which English preposition fits best.

What exactly does akwai mean, and is it like “there is/there are”?

Yes. Akwai is the Hausa existential verb, used to say that something exists or is present somewhere.
In English you say “there is / there are”; in Hausa you just use akwai for both singular and plural:

Akwai mutum. – There is a person.
Akwai mutane. – There are people.

So in the sentence, akwai expresses “there are”.

Does akwai change for singular vs. plural, or past vs. present?

For number, akwai stays the same: it is used with both singular and plural.

For tense:

Present/general: akwai – there is / there are
Past (often): akwai can still be used, but many speakers use akwai with time words or ya kasance / suna akwai in more formal speech, or simply rely on context.

For basic learner usage, you can treat akwai as an unchanging “there is/are” verb in the present or general sense.

What does mutane mean, and is it related to mutum?

Mutane means “people”.
It is the plural of mutum (“person, human being”).

mutum – a person
mutane – people

This is an irregular plural; you don’t just add a suffix. You have to learn mutum → mutane as a pair.

What is the function of masu in mutane masu yawa?

Masu is a particle used to form a kind of descriptive phrase, often meaning “those who have/are characterized by …”.
In mutane masu yawa:

mutane – people
masu – those who have / people who are
yawa – much / a lot / many-ness

Literally it’s like “people who have much/many”, but idiomatically it just means “many people”.
This mutane masu X pattern is used often, e.g. mutane masu kudi – rich people (people with money).

Why is it masu and not mai in this sentence?

Mai and masu are related forms:

mai is used with singular nouns: one who has / owner of / characterized by
masu is used with plural nouns: those who have / ones who are characterized by

Examples:

mutum mai kudi – a rich person (person with money)
mutane masu kudi – rich people (people with money)

Since mutane is plural, you must use masu, giving mutane masu yawa (“many people”).

Is there a difference between mutane masu yawa and mutane da yawa?

Both can mean “many people”, but there is a slight nuance:

mutane masu yawa – literally “people who have much”; a bit more “descriptive” or adjective-like.
mutane da yawa – literally “people with many / people of many”; very common everyday way to say “many people”.

In everyday speech, they often overlap in meaning. You will hear mutane da yawa very frequently; mutane masu yawa is also correct and natural.

How does yawa work here – is it an adjective like “many” or a noun?

Yawa is originally more like a noun meaning “abundance, plenty, much-ness”, but in combinations like da yawa or masu yawa, it functions like “many / much / a lot”.

Common patterns:

mutane da yawa – many people
abinci da yawa – a lot of food
mutane masu yawa – many people (people characterized by abundance)

So even though yawa is a noun historically, you can think of it as the “many / a lot” element in these phrases.

Why doesn’t the sentence use something like suna or suke to say “they are”, as in mutane suna yawa?

In Hausa, akwai already expresses existence/presence (“there are”), so you don’t need an extra “they are” verb here.

A kasuwa akwai mutane masu yawa. – At the market there are many people.

A sentence like mutane suna yawa would mean something more like “people are becoming many” (increasing in number), which is a different idea.
So for simply stating that many people are present somewhere, akwai is the natural verb to use.

Do I need a word for “the” in A kasuwa to say “at the market”, or is kasuwa already definite?

Hausa does not use a separate word for “the” like English does.
Kasuwa by itself can mean “a market” or “the market”, depending on context.

Ina zuwa kasuwa. – I’m going to (the) market.
A kasuwa akwai mutane masu yawa. – At the market there are many people.

Listeners understand from context whether you mean a specific market or just “market” in general.

Could the sentence be Akwai mutane masu yawa a kasuwa instead, and is there any difference?

Yes, Akwai mutane masu yawa a kasuwa is fully correct and very common.

The difference is mainly focus/order:

A kasuwa akwai mutane masu yawa. – Starts with the place: “At the market, there are many people.” (location-focused)
Akwai mutane masu yawa a kasuwa. – Starts with existence/people: “There are many people at the market.” (existence/people-focused)

Meaning is essentially the same; the choice depends on what you want to emphasize.

Is there any difference between a kasuwa and something like a cikin kasuwa?

Yes:

a kasuwa – at/in the market (general location; can be in or around the market area)
a cikin kasuwa – inside the market, more specifically inside the market space/enclosure.

So:

A kasuwa akwai mutane masu yawa. – At the market there are many people.
A cikin kasuwa akwai mutane masu yawa. – Inside the market there are many people.

Both are correct; a cikin is just more explicit about “inside”.