Comme l’entrée était sale, ma mère a repris la serpillière et a rempli le seau une deuxième fois.

Yes, comme often means like or as, but here it has a different job: it introduces a reason.

At the beginning of a sentence, comme can mean since, as, or because:

Comme l’entrée était sale, ... = Since/As the entrance was dirty, ...

This use is very common when the reason is background information.
A similar sentence with parce que is possible, but the structure is usually different:

Ma mère a repris la serpillière parce que l’entrée était sale.

So in this sentence, comme is a causal conjunction, not a comparison word.

Because entrée begins with a vowel sound, la shortens to l’. This is called elision.

So:

• la table
• la porte
• l’entrée

The noun entrée is feminine, which is why the full article would be la if elision were not needed.

This is a classic French tense contrast:

• était = imparfait
• a repris, a rempli = passé composé

The imparfait is used for background, description, or a state:

• l’entrée était sale describes the situation

The passé composé is used for completed actions:

• ma mère a repris la serpillière
• a rempli le seau une deuxième fois

So the sentence is structured like this:

• background: the entrance was dirty
• main actions: she picked up the mop again and filled the bucket again

This is one of the most common reasons French switches between these two past tenses in the same sentence.

Because these verbs use avoir as the auxiliary.

With avoir, the past participle usually does not agree with the subject. So even though ma mère is feminine, you still get:

• a repris
• a rempli

Agreement with avoir only happens if a direct object comes before the participle.

Here, the direct objects come after:

• a repris la serpillière
• a rempli le seau

So there is no agreement.

Here reprendre means to take again, pick up again, or resume using.

So a repris la serpillière suggests that the mother had already been cleaning, stopped, noticed it was still dirty, and then took the mop again.

It does not mean took back in the sense of recovering something from another person.
It is more like she picked the mop back up.

Also, reprendre is irregular:

• infinitive: reprendre
• past participle: repris

That is why it is a repris, not something like a reprendu.

Because French often repeats the auxiliary when two past actions are linked by et:

ma mère a repris la serpillière et a rempli le seau

This is completely normal and very clear.

Sometimes French can omit the second auxiliary when the subject is the same, especially in more compact writing:

ma mère a repris la serpillière et rempli le seau

But for learners, the full version with both a forms is easier to understand and very natural.

French often uses the definite article when the object is already understood from the context.

Here, the mop and bucket are the obvious cleaning tools involved in the situation, so la serpillière and le seau sound natural.

Using sa serpillière would emphasize ownership:

• her mop

But in ordinary narration, French often prefers the simple definite article when ownership is not the important point.

Une deuxième fois means a second time or once more.

It focuses on the fact that the action happened again:

• she filled the bucket one more time

Deux fois usually means twice in a more general counting sense.

So the nuance is:

• deux fois = twice
• une deuxième fois = a second time / again

In this sentence, une deuxième fois fits well because it highlights repetition in the story.

Yes, pour la deuxième fois is possible, but it is a little more explicit and slightly heavier in tone.

Compare:

• a rempli le seau une deuxième fois = very natural, smooth narrative style
• a rempli le seau pour la deuxième fois = also correct, but more pointed: for the second time

In everyday storytelling, une deuxième fois is often the more natural choice.

Because the sentence begins with a subordinate clause:

Comme l’entrée était sale, ...

That opening clause gives the reason, and the comma separates it from the main clause:

ma mère a repris la serpillière et a rempli le seau une deuxième fois

This is similar to English punctuation when a dependent clause comes first. It helps the sentence read more clearly.