Mi prima se esfuerza mucho, y por eso cada semana habla con más confianza.

Because the usual verb here is esforzarse, a pronominal verb meaning to make an effort or to try hard.

So:

• me esfuerzo = I make an effort
• te esfuerzas = you make an effort
• se esfuerza = she/he makes an effort

In this sentence, se is part of the verb, not a separate word meaning herself in a literal way.

It is:

• third person singular
• present tense
• from esforzarse

So mi prima se esfuerza means my cousin tries hard / makes a big effort.

The subject is mi prima, so the verb must be se esfuerza.

Because mucho modifies a verb here, while muy usually modifies an adjective or adverb.

• se esfuerza mucho = she tries hard / she makes a big effort
• muy difícil = very difficult
• muy bien = very well

A simple rule:

• mucho = a lot
• muy = very

So se esfuerza mucho is correct.

Por eso means because of that, that’s why, or for that reason.

It links the two ideas:

• she works hard
• as a result, she speaks with more confidence each week

So y por eso is something like and that’s why or and because of that.

Both can mean every week.

• cada semana = every week
• todas las semanas = every week / all weeks

In this sentence, cada semana is a very natural, compact way to express regular repetition. After cada, Spanish normally uses a singular noun:

• cada día
• cada mes
• cada semana

Spanish often uses the present tense for things that happen regularly or habitually.

Here, cada semana shows that this is a repeated action, so the present is natural:

• cada semana habla con más confianza = each week she speaks with more confidence

It does not mean only right now. It can also describe a pattern or ongoing development.

Because con + noun is often the most natural way in Spanish to express this idea.

• habla con más confianza = she speaks with more confidence

Although a form like más confiadamente might be understandable, it sounds much less natural in everyday Spanish. Native speakers very often prefer:

• con confianza
• con más seguridad
• con más fluidez

So con más confianza is a very normal Spanish way to say it.

It is comparing her current way of speaking with how she spoke before.

Spanish often leaves that comparison implicit. So:

• habla con más confianza = she speaks with more confidence
• understood meaning: more confidence than before

You do not need to add the second part if the context is already clear.

Yes, confianza can mean different but related things depending on context:

• trust
• confidence
• self-confidence
• sometimes even familiarity

In this sentence, because the topic is how she speaks, confianza means confidence/self-confidence.

So habla con más confianza means she speaks more confidently or with greater self-assurance.

In modern standard Spanish, possessives like mi, tu, su, nuestro, etc. normally go directly before the noun without an article.

So:

• mi prima = my cousin
• tu hermano = your brother
• nuestra casa = our house

English speakers sometimes expect an article because some older forms or regional varieties use one, but in normal modern Spanish from Spain, mi prima is the standard form.

Yes.

• primo = male cousin
• prima = female cousin

So mi prima tells you that the cousin is female.

Not always. In many cases, Spanish does not use a comma before y when simply joining two clauses.

So you could also write:

Mi prima se esfuerza mucho y por eso cada semana habla con más confianza.

The comma in your sentence can be understood as marking a pause, but it is not strictly necessary for the basic meaning.

In standard Peninsular Spanish:

• esfuerza is roughly es-FWER-tha

Important points:

• ue forms a diphthong: fuer
• the z in Spain is usually pronounced like the th in think
• the stress falls on fuer

So se esfuerza sounds roughly like seh es-FWER-tha.

Yes. Spanish word order is flexible, especially with time expressions.

For example, these are all natural:

• Mi prima se esfuerza mucho, y por eso cada semana habla con más confianza.
• Cada semana, mi prima habla con más confianza porque se esfuerza mucho.
• Mi prima habla cada semana con más confianza.

The original sentence is clear and natural, but Spanish often lets you move elements like cada semana for emphasis or style.