После трудного дня мне нужна поддержка друга.

Because после always requires the genitive case.

• Dictionary form: трудный день = a difficult day
• After после, both words change to genitive:
  • трудный → трудного
  • день → дня

So:

• после трудного дня = after a difficult day

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work
• после дождя = after the rain

Russian often expresses need differently from English.

The structure here is:

• мне нужна поддержка
  literally: to me is needed support

So мне is in the dative case, not the nominative я, because Russian is not saying I need... word-for-word. It is saying something more like:

• Support is necessary for me
• I am in need of support

This pattern is very common:

• мне нужен совет = I need advice
• мне нужна помощь = I need help
• мне нужно время = I need time

You can also say:

• Я нуждаюсь в поддержке = I need support / I am in need of support

But that sounds a bit more formal or bookish in many situations.
Мне нужна поддержка is a very natural everyday way to say it.

Because нужна agrees with поддержка, and поддержка is a feminine noun.

In this pattern, нужен / нужна / нужно / нужны changes to match the thing needed:

• нужен for masculine nouns
  
  • мне нужен друг = I need a friend
• нужна for feminine nouns
  
  • мне нужна поддержка = I need support
• нужно for neuter nouns
  
  • мне нужно время = I need time
• нужны for plural nouns
  
  • мне нужны деньги = I need money

So the ending -а in нужна is there because поддержка is feminine singular.

Because поддержка is the grammatical subject of the sentence.

Even though English says I need support, Russian structures this as:

• мне нужна поддержка
• literally: to me, support is needed

In that structure, поддержка is the thing that is needed, so it stays in the nominative case.

Compare:

• Я вижу поддержку = I see the support
  Here поддержку is accusative because it is a direct object.

But in:

• Мне нужна поддержка
  поддержка is not a direct object. It is the subject-like element that matches нужна.

Друга is the genitive singular of друг.

Here it depends on поддержка:

• поддержка друга = a friend’s support / support from a friend

So the pattern is:

• поддержка кого? = support of whom?
• answer: друга

This is a very common noun + genitive pattern in Russian:

• совет друга = a friend’s advice
• помощь сестры = a sister’s help
• слова учителя = the teacher’s words

In this sentence, it normally means:

• a friend’s support
• support from a friend

That is the most natural interpretation.

Russian genitive after nouns can sometimes be broad, but here поддержка друга would usually be understood as support coming from the friend, not support for the friend.

If you specifically wanted to say support for a friend, Russian would usually express that differently, for example:

• поддержка для друга = support for a friend
• поддержка друга в трудный момент could still be ambiguous without context, but usually means the friend’s support

So in your sentence, learners should understand друга as the source/owner of the support.

By itself, друга means of a friend or from a friend. It does not automatically mean my friend.

So:

• поддержка друга = support of a friend / support from a friend
• поддержка моего друга = support of my friend / support from my friend

However, in real conversation, Russian often leaves things less specific if the context is clear. So depending on the situation, поддержка друга might be understood as support from a friend in a general sense.

If you want to be explicit, you can say:

• мне нужна поддержка моего друга = I need my friend’s support
• мне нужна поддержка друзей = I need support from friends

Yes, Russian word order is flexible, though different orders sound slightly different in emphasis.

The neutral version here is:

• После трудного дня мне нужна поддержка друга.

This is very natural. It sets the scene first: After a difficult day...

You could also say:

• Мне нужна поддержка друга после трудного дня.
• После трудного дня поддержка друга мне нужна.

These are grammatically possible, but they shift emphasis.

Very roughly:

• После трудного дня мне нужна поддержка друга
  neutral, natural
• Мне нужна поддержка друга после трудного дня
  stronger focus on I need
• Поддержка друга мне нужна
  stronger focus on a friend’s support

Russian uses word order partly for information structure, not just grammar.

Нужна is not a verb. It is the short form of the adjective нужный = needed / necessary.

Short forms are often used in Russian in predicate sentences like this:

• нужен
• нужна
• нужно
• нужны

So:

• мне нужна поддержка
  literally: support is necessary/needed for me

This is why it changes for gender and number.

You do not have to memorize all the theory right away, but it helps to know that нужен / нужна / нужно / нужны behaves more like an adjective than a normal verb.

Yes. That is correct.

• Мне нужна поддержка друга = I need a friend’s support
• Мне необходима поддержка друга = I need a friend’s support / A friend’s support is necessary for me

The version with необходима sounds a bit more formal, stronger, or more serious.

Compare:

• нужна = everyday, common, natural
• необходима = more formal, emphatic, sometimes more dramatic

So for ordinary speech, нужна is usually the best choice.

A rough pronunciation guide is:

• После трудного дня мне нужна поддержка друга
• POS-lye trood-na-va dnya mnye noozh-NA pat-DYERZH-ka DROO-ga

A few useful points:

• после sounds roughly like POS-lye
• трудного is pronounced closer to trood-na-va than you might expect from the spelling
• мне is one syllable: mnye
• нужна has the stress on the last syllable: noozh-NA
• поддержка has the stress on -держ-: pat-DYERZH-ka
• друга has the stress on the first syllable: DROO-ga

Stress is important in Russian, so it helps to learn words together with stress:

• после
• тру́дного
• дня
• мне
• нужна́
• подде́ржка
• дру́га

Yes, it is natural and correct.

It sounds like a thoughtful, emotionally expressive sentence. A Russian speaker could definitely say it.

Depending on style and context, other natural variants are also possible:

• После трудного дня мне нужна поддержка.
  After a difficult day, I need support.
• После тяжёлого дня мне нужна поддержка друга.
  After a hard day, I need a friend’s support.
• После трудного дня мне очень нужна поддержка друга.
  After a difficult day, I really need a friend’s support.

So your original sentence is fully acceptable and natural.