Moja kći nosi inhalator sa sobom, jer joj pomaže kad kašlje noću.

Because kći is a feminine noun, and the possessive adjective must agree with it in gender, number, and case. The subject here is singular feminine nominative, so the correct form is moja.

You can compare: moj sin = my son
moja kći = my daughter
moje dijete = my child

It is in the nominative singular because it is the subject of the sentence: the daughter is the one doing the action of nosi.

Kći is also an irregular noun, so learners often notice that it does not behave as simply as many other feminine nouns.

Croatian does not have articles like English a or the. So inhalator can mean an inhaler or the inhaler, depending on context.

That means Croatian learners of English have to add articles, and English speakers learning Croatian have to get used to leaving them out.

Nosi is the 3rd person singular present tense of nositi. In this sentence it means carries.

The verb nositi can also mean to wear, depending on context: nosi torbu = she carries a bag
nosi kaput = she is wearing a coat

Here, because the object is inhalator, the natural meaning is carries.

Sa sobom means with herself / with her in a reflexive sense. It is used because the person carrying the inhaler and the person the inhaler is with are the same person: the daughter.

So: nosi inhalator sa sobom = she carries the inhaler with her

The word sobom is the instrumental form of the reflexive pronoun sebe.

Because Croatian normally uses the reflexive pronoun when the person mentioned is the same as the subject of the clause.

So: Moja kći nosi inhalator sa sobom = my daughter carries the inhaler with her herself

If you said s njom, it would usually suggest with her as a non-reflexive pronoun, often referring to some other female person, or it would sound less standard in this context.

Sa is a variant of s used for easier pronunciation. It is especially common before words beginning with s, š, z, ž, or certain consonant clusters.

Since sobom begins with s, sa sobom sounds smoother than s sobom.

Because the verb pomagati takes the dative, not the accusative.

So: pomagati komu? čemu? = to help whom? what?
joj = to her

By contrast, ju is accusative and would be used for a direct object, not after pomagati.

In English we simply say it helps her, but in Croatian that her is dative: joj.

Because joj is a clitic. Croatian clitics usually appear near the beginning of the clause, often in the so-called second position.

So jer joj pomaže is the natural order.
Jer pomaže joj sounds wrong or very unnatural in standard Croatian.

This is one of the big differences from English word order.

The subject is understood from context: inhalator. So pomaže means it helps.

Croatian often leaves subject pronouns unstated when the verb ending already shows the person and number, and when the context makes the subject clear.

English needs it helps her, but Croatian can simply say joj pomaže.

Both mean when. Kad is the shorter form and is extremely common in everyday speech and writing. Kada is slightly fuller and can sound a bit more formal or emphatic.

So in this sentence, kad kašlje is completely natural: kad = when
kada = when

Kašlje is the 3rd person singular present tense of kašljati = to cough.

So: ja kašljem = I cough
ti kašlješ = you cough
ona kašlje = she coughs

Here it matches kći, so kad kašlje means when she coughs.

Noću means at night or during the night. In modern Croatian, it is best learned as a common adverbial time expression.

So: kašlje noću = she coughs at night

Learners do not need to analyze it too deeply at first; it is very useful to memorize as a whole expression.

Because jer introduces a subordinate clause meaning because, and Croatian normally separates that clause with a comma.

So: Moja kći nosi inhalator sa sobom, jer joj pomaže...

This is standard Croatian punctuation.

Yes, Croatian word order is more flexible than English, but not all changes sound equally natural. The given sentence is neutral and natural.

For example, these are possible with different emphasis: Moja kći nosi inhalator sa sobom...
Moja kći sa sobom nosi inhalator...

But the clitic joj still has to stay in its usual early position: jer joj pomaže is natural
jer pomaže joj is not

This is a word many learners find difficult. It is one syllable, and the ć is a soft sound, softer than English ch.

A rough English-friendly approximation is something like k-chee or k-tyee, but neither is exact. The important thing is that ć is softer than č, and i is pronounced like ee in see.