На десерт мы взяли по кусочку шоколадного торта.

На десерт means for dessert.

Literally, на often means on/to/for, but in many set expressions it has to be translated more naturally. In Russian, на + accusative is commonly used to mean something like for use as, for, or as part of an occasion:

• на завтрак — for breakfast
• на обед — for lunch
• на ужин — for dinner
• на десерт — for dessert

So На десерт мы взяли... is idiomatic Russian for For dessert, we took/ordered...

Because this is на + accusative, not на + prepositional.

Compare:

• на десерт — for dessert → accusative
• о десерте — about the dessert → prepositional
• на десерте would usually mean on the dessert, as in physically on top of it

So here Russian is not talking about location; it is expressing purpose or role: for dessert.

Because the subject is мы = we.

Russian past tense agrees with the subject in gender/number:

• я взял / я взяла — I took
• он взял — he took
• она взяла — she took
• они / мы взяли — they/we took

So мы взяли = we took.

This is an aspect question.

• взять / взяли = perfective
• брать / брали = imperfective

Here the speaker is referring to a completed action: they made a choice and took/ordered the cake. That is why взяли is natural.

Very roughly:

• Мы брали торт — we were taking / used to take / took (with focus on process or repeated action)
• Мы взяли торт — we took / ordered the cake (completed event)

In restaurant or food contexts, взять often means to take / to get / to order.

По кусочку means a piece each or one little piece each.

This is a very common Russian pattern:

• по
  • a noun in the dative
• it often means one each, apiece, or in portions

So:

• по яблоку — one apple each
• по чашке чая — a cup of tea each
• по кусочку торта — a piece of cake each

In this sentence, мы взяли по кусочку шоколадного торта means that each person took their own piece.

Because after по, Russian often uses the dative case in this kind of distributive meaning.

The base noun is:

• кусочек — a little piece

Its dative singular is:

• кусочку

So:

• по кусочку = a little piece each

This is not the normal dictionary form, but a case form required by по.

Кусок means a piece / a chunk, while кусочек is a diminutive form, meaning a little piece or a small piece.

So:

• кусок торта — a piece of cake
• кусочек торта — a small piece / a little piece of cake

The diminutive can suggest:

• small size
• affection
• politeness
• a softer, less blunt tone

In food contexts, diminutives are very common and natural in Russian.

Because кусочек is followed by the thing of which it is a piece, and that thing goes into the genitive:

• кусочек чего? — a piece of what?
• кусочек торта — a piece of cake

Since торта is masculine singular genitive, its adjective must match it:

• nominative: шоколадный торт
• genitive: шоколадного торта

So:

• кусочек шоколадного торта = a small piece of chocolate cake

Because the sentence is not saying a chocolate cake as a whole. It is saying a piece of chocolate cake.

After words like кусок, кусочек, чашка, бутылка, стакан, Russian usually puts the following noun in the genitive:

• чашка чая — a cup of tea
• бутылка воды — a bottle of water
• кусок хлеба — a piece of bread
• кусочек шоколадного торта — a small piece of chocolate cake

So шоколадный торт would be nominative, but here genitive is required.

In this context, it often means something like took, got, or ordered, depending on the situation.

Russian commonly uses взять in everyday speech for choosing food or drink:

• Я взял кофе. — I got / had / ordered a coffee.
• Мы взяли торт. — We got / ordered cake.

So the exact English translation depends on context, but the Russian is perfectly normal.

Yes, it could be omitted.

Russian often leaves out subject pronouns when they are clear from the verb:

• На десерт взяли по кусочку шоколадного торта.

This still strongly suggests we took... if the context makes it clear.

However, мы may be included:

• for clarity
• for emphasis
• to contrast with someone else

For example:

• Они взяли мороженое, а мы взяли по кусочку шоколадного торта.
  They got ice cream, and we took a piece of chocolate cake each.

Yes, Russian word order is flexible.

На десерт мы взяли по кусочку шоколадного торта is natural because it starts with the topic for dessert, then gives the subject and action.

Other possible orders include:

• Мы на десерт взяли по кусочку шоколадного торта.
• Мы взяли на десерт по кусочку шоколадного торта.

These are all grammatical. The difference is mainly in focus and emphasis:

• На десерт first → highlights that this is what they had for dessert
• Мы first → highlights we
• по кусочку шоколадного торта later → keeps the new information toward the end

They are very close, but по одному кусочку is more explicit.

• по кусочку — a piece each
• по одному кусочку — one piece each

Usually по кусочку already implies one each, so adding одному is often unnecessary unless you want to stress the number one.

Yes, very possibly.

Because of кусочку, the sentence can suggest:

• one piece each
• and also that each piece was small

So there is a slight nuance of moderation or modesty: we each had a little piece of chocolate cake.

That nuance comes from the diminutive -очек in кусочек.