Zamolit ću učiteljicu da mi još jednom objasni ovaj zadatak.

Both patterns are possible in Croatian.

• Zamolit ću = the auxiliary ću comes after the infinitive, and in standard spelling the final -i of zamoliti is dropped.
• Ja ću zamoliti = the auxiliary comes before the infinitive, so the full infinitive stays.

So these are both correct:

• Zamolit ću učiteljicu...
• Ja ću zamoliti učiteljicu...

The first one is very common and natural.

It is the future tense: I will ask.

Croatian future is made with:

• the infinitive of the main verb
• plus forms of htjeti used as an auxiliary: ću, ćeš, će, ćemo, ćete, će

Here:

• zamoliti = to ask/request
• ću = I will

So zamolit ću means I will ask.

Because it is in the accusative case.

The verb zamoliti takes a direct object: you ask someone.
Here, the person being asked is učiteljica → učiteljicu in the accusative singular.

So:

• učiteljica = nominative, the teacher
• učiteljicu = accusative, the teacher as the object of the verb

This is a question of aspect.

• zamoliti is perfective: to ask once, to make a single request
• moliti is imperfective: to ask/beg, to be asking, or to ask repeatedly

In this sentence, the speaker means one complete action: I will ask the teacher.
That is why zamoliti fits better.

After verbs like zamoliti, Croatian usually uses da + present tense where English often uses an infinitive.

So English says:

• I will ask the teacher to explain...

Croatian says:

• Zamolit ću učiteljicu da objasni...

Literally, this is closer to I will ask the teacher that she explain..., although that is not natural English. It is just the normal Croatian structure.

Because after da, Croatian often uses a present-tense form to talk about a desired, expected, or future action.

Here objasni is the 3rd person singular form of objasniti:

• da ... objasni = that she explain / to explain

It refers to what the teacher is supposed to do after being asked. So even though the form is present, the meaning is future-looking in this context.

Because its subject is učiteljica — the teacher — which is singular.

So the clause means:

• da mi još jednom objasni ovaj zadatak
• that she explain this task to me one more time

The verb agrees with učiteljica, not with the speaker.

Mi here means to me.

The verb objasniti often works like this:

• objasniti komu što = to explain something to someone

So:

• mi = to me (dative clitic)
• ovaj zadatak = this task (the thing being explained)

Why not the other forms?

• mene = me, usually accusative or genitive, so it does not fit here
• meni = the full stressed dative form, used for emphasis or contrast
• mi = the normal unstressed form, which is the natural choice here

It means one more time or once again.

• još = still, more, yet
• jednom = once / one time

Together, još jednom means that the speaker wants the teacher to explain the task again.

Because ovaj zadatak is the direct object of objasni, so it is in the accusative case.

But zadatak is a masculine inanimate noun, and for masculine inanimate nouns, the accusative singular looks the same as the nominative singular.

So:

• nominative: ovaj zadatak
• accusative: ovaj zadatak

That is why the form does not change.

Because Croatian often drops subject pronouns when they are not needed.

The form ću already shows that the subject is I. So ja is unnecessary unless you want emphasis.

Compare:

• Zamolit ću učiteljicu... = normal, neutral
• Ja ću zamoliti učiteljicu... = I will ask the teacher, with extra emphasis

Yes, Croatian word order is fairly flexible, but not completely free.

For example, these are possible:

• Zamolit ću učiteljicu da mi još jednom objasni ovaj zadatak.
• Učiteljicu ću zamoliti da mi još jednom objasni ovaj zadatak.

The meaning stays basically the same, but the emphasis changes slightly.

However, small words like ću and mi are clitics, so they usually have to stand near the beginning of their clause. That is why their position is more restricted than the position of nouns like učiteljicu or ovaj zadatak.