На десерт мы взяли по кусочку арбуза, а детям дали по одной сливе.

Here по has a distributive meaning: one each, apiece, per person.

So:

• мы взяли по кусочку арбуза = we took a piece of watermelon each
• детям дали по одной сливе = the children were given one plum each

This is a very common use of по in Russian.

Because after по in this distributive meaning, Russian commonly uses the dative.

So:

• по кусочку
• по одной сливе

Both nouns are in the dative singular.

This is one of the standard patterns learners memorize:

• по + dative = one each / apiece / per

Examples:

• дать по яблоку = to give one apple each
• взять по билету = to take one ticket each

Because кусочек means a piece / a little piece, and the thing of which you take a piece usually goes into the genitive.

So:

• кусочек чего? — арбуза

This is similar to English a piece of watermelon.

Structure:

• по кусочку арбуза
  • по кусочку = one little piece each
  • арбуза = of watermelon

So the dative is caused by по, but the genitive арбуза is caused by кусочек.

Детям is the dative plural of дети and marks the recipient: the children are the ones who received the plums.

In Russian, the person something is given to is usually in the dative:

• дать детям = to give to the children
• сказать маме = to say to mother
• помочь другу = to help a friend

So in:

• а детям дали по одной сливе

the children are the indirect object: they gave the children one plum each.

There are two reasons.

1. Agreement Один must agree with слива in gender, number, and case.

Since слива is feminine singular, one starts from the feminine form:

• nominative: одна слива

2. Case Because of по, the phrase is in the dative, so both words change:

• одна слива → по одной сливе

So:

• одной = feminine dative singular of один
• сливе = dative singular of слива

Кусочек is a diminutive form of кусок.

• кусок = a piece
• кусочек = a little piece, a small piece

In everyday speech, diminutives like this often sound more natural, friendly, or simply more idiomatic, especially when talking about food.

So по кусочку арбуза suggests a little piece of watermelon each.

It does not always have to sound cute; often it just sounds natural.

Here а means something like and, but more specifically it often marks a contrast or switch of focus.

So the sentence is roughly:

• For dessert, we took a piece of watermelon each, and the children were given one plum each.

But а suggests a slight contrast:

• we got one thing
• the children got another

Russian often uses а where English might use:

• and
• while
• whereas
• but

depending on context.

Both are possible, but starting with На десерт puts the setting or context first:

• As for dessert...
• For dessert...

Russian word order is more flexible than English word order, and it is often used to highlight what the speaker wants to frame first.

So:

• На десерт мы взяли... = emphasis on for dessert
• Мы взяли на десерт... = a bit more neutral in structure

Both are natural.

Because both verbs refer to plural subjects.

• мы взяли = we took
• дали implies a plural subject such as they gave or we gave, depending on context

Russian past tense agrees in gender/number, not person.

So:

• masculine singular: взял
• feminine singular: взяла
• neuter singular: взяло
• plural: взяли

And similarly:

• дал
• дала
• дало
• дали

Russian often omits the subject when it is clear from context or not important.

So дали can mean:

• they gave
• we gave
• someone gave

depending on the situation.

In this sentence, the important information is not really who gave the plums, but that the children received one plum each.

This kind of omitted subject is very normal in Russian.

Yes, it is a very common expression meaning for dessert.

Examples:

• На десерт был торт. = For dessert there was cake.
• Что у нас на десерт? = What do we have for dessert?
• На десерт дали фрукты. = They served fruit for dessert.

So in your sentence, На десерт sets the scene: this happened as part of the dessert course.