Перед поїздкою дочка обіймає матір, а мати каже, що теж сумуватиме за нею.

Після is not being used here; the phrase is перед поїздкою, meaning before the trip.

The noun поїздка changes to поїздкою because перед normally takes the instrumental case when it means before or in front of.

• поїздка = trip, यात्रा / journey
• поїздкою = instrumental singular

So:

• перед поїздкою = before the trip

A learner may also see до поїздки in other contexts. That can also mean before the trip, but перед поїздкою often feels more like right before / in connection with the upcoming trip.

Both дочка and донька mean daughter.

In modern Ukrainian, many speakers feel that донька is a bit more everyday or natural in conversation, but дочка is also fully standard and common. In this sentence, there is no important difference in meaning.

So for a learner, the main thing is:

• дочка = daughter
• донька = daughter

You should recognize both.

They are two forms of the same noun:

• мати = mother, in the nominative case
• матір = mother, in the accusative case

In the sentence:

• дочка обіймає матір = the daughter hugs the mother
  Here матір is the object, so accusative is used.
• мати каже = the mother says
  Here мати is the subject, so nominative is used.

This noun is somewhat irregular, so its forms do not look as predictable as many other feminine nouns.

A few useful forms are:

• мати — nominative
• матері — genitive/dative
• матір — accusative
• матір’ю — instrumental

Here it is the noun mother.

That can confuse learners because мати is also the infinitive of the verb to have.

In this sentence, though, the structure makes it clear:

• мати каже = the mother says

Here мати is the subject of каже.

If it were the verb to have, the sentence would be built differently.

So context is very important:
мати can mean either mother or to have, but here it clearly means mother.

Обіймає is the 3rd person singular present form of обіймати.

It can mean either:

• hugs
• is hugging

Ukrainian does not usually make the same strict distinction that English does between simple present and present continuous. A present-tense form like обіймає can describe an action happening right now.

So in a scene description, дочка обіймає матір is perfectly natural for the daughter is hugging her mother.

Also, обіймати is imperfective, which fits an ongoing action.
Its perfective partner is обійняти. A perfective verb normally does not have a true present meaning in this kind of sentence.

In Ukrainian, а often marks a change of subject, contrast, or shift in focus.

Here the sentence moves from one person’s action to another person’s speech:

• дочка обіймає матір
• а мати каже...

So а works well because it signals something like:

• and
• while
• whereas
• sometimes even a mild but

In this sentence, it is not a strong contradiction. It is more of a transition from the daughter to the mother.

If you used і, it would sound more like simple addition.
Using а is more natural here.

Both commas are required by normal Ukrainian punctuation.

1. Before а
   There are two coordinated clauses:

   • дочка обіймає матір
   • мати каже...

   So a comma is placed before а.

2. Before що
   Що introduces a subordinate clause:

   • мати каже, що теж сумуватиме за нею

   So a comma is placed before що.

This is very standard in Ukrainian. Even if English punctuation might look different, Ukrainian normally keeps these commas.

Що here means that.

It introduces what the mother says:

• мати каже, що... = the mother says that...

So the structure is:

• main clause: мати каже
• subordinate clause: що теж сумуватиме за нею

English often allows that to be omitted:

• The mother says she will also miss her

But in Ukrainian, що is usually stated explicitly in this kind of sentence.

Теж means also or too.

In this sentence:

• що теж сумуватиме за нею

it means that the mother will also miss her.

The placement of теж affects emphasis. Here it naturally modifies the idea of missing her too.

Compare:

• мати каже, що теж сумуватиме за нею
  = the mother says that she will also miss her

• мати теж каже, що сумуватиме за нею
  = the mother also says that she will miss her

The difference is subtle, but the focus changes.
In the original sentence, теж is most naturally understood as applying to сумуватиме.

Сумуватиме is the 3rd person singular future of сумувати.

The infinitive is:

• сумувати = to miss, to feel sad, to long for

The form:

• сумуватиме = he/she will miss

This is a normal future form of an imperfective verb in Ukrainian.

Ukrainian can often express the future of an imperfective verb in two ways:

• сумуватиме
• буде сумувати

Both can mean will miss.

The synthetic form сумуватиме is compact and very natural.
So this sentence could also be phrased with буде сумувати, but сумуватиме is completely standard.

This is because the verb сумувати uses the pattern:

• сумувати за ким? за чим?

So when Ukrainian says to miss someone, it does not use a direct object the way English does. Instead, it uses за + instrumental.

Here нею is the instrumental singular form of вона:

• вона = she
• нею = her, in the instrumental

So:

• сумуватиме за нею = will miss her

This is a very important pattern to remember:

• сумувати за мамою = miss mom
• сумувати за другом = miss a friend
• сумувати за нею = miss her

Yes. Ukrainian word order is fairly flexible because case endings show grammatical roles.

The original order:

• Перед поїздкою дочка обіймає матір, а мати каже, що теж сумуватиме за нею.

is natural and gives this flow:

1. set the time frame: Перед поїздкою
2. describe the daughter’s action
3. shift to the mother’s response

You could also say:

• Дочка обіймає матір перед поїздкою...

That is also correct, but the emphasis changes slightly.
The original sentence highlights before the trip right away.

So the word order is not random: it helps show focus, topic, and what the speaker wants to foreground.