Пусть мандарин и маленький, после ужина мне иногда хочется именно его, а не большой десерт.

Here пусть does not mean let. In this sentence, пусть ... и ... is a set pattern meaning:

• although ...
• even if ...
• granted that ...

So:

• Пусть мандарин и маленький = Although the mandarin is small / The mandarin may be small, but...

The little и is part of this concessive pattern. It helps create the meaning “yes, that may be true, but nevertheless...”

A few similar examples:

• Пусть он и молодой, но очень опытный.
  He may be young, but he is very experienced.

• Пусть это и дорого, зато качественно.
  It may be expensive, but it’s high quality.

So in your sentence, the speaker is acknowledging one fact—the mandarin is small—before saying they still want it.

Because мандарин is the subject of the concessive clause:

• Пусть мандарин и маленький
  literally: Although the mandarin is small

The subject is in the nominative case, so we get мандарин.

If it were in another role, the case would change. For example:

• У меня нет мандарина. — I don’t have a mandarin.
• Я ем мандарин. — I’m eating a mandarin.

But here, the idea is essentially мандарин маленький = the mandarin is small, so nominative is correct.

Because маленький is a predicate adjective describing the subject мандарин.

The underlying structure is:

• Мандарин маленький.
  The mandarin is small.

In Russian, in the present tense, the verb быть (to be) is usually omitted, so you just say:

• мандарин маленький

Since мандарин is masculine singular nominative, the adjective also appears as маленький.

You would use other forms in other constructions:

• маленьким — instrumental, often after verbs like быть in past/future or in some other structures
• маленького — genitive/accusative masculine animate, depending on context

But none of those are needed here. This is just a plain present-tense description: the mandarin is small.

Because после requires the genitive case.

So:

• после завтрака — after breakfast
• после обеда — after lunch
• после ужина — after dinner

The noun ужин becomes ужина in the genitive singular.

This is just something you have to learn with the preposition:

• после + genitive

Both are possible, but they are not exactly the same.

Я хочу

This is the straightforward, direct verb I want.

• Я хочу мандарин.
  I want a mandarin.

Мне хочется

This is a very common impersonal construction meaning something like:

• I feel like
• I have a desire for
• I’m in the mood for

Literally, it works more like “to me, it is wanted”, which is why мне is in the dative case.

So:

• мне хочется sounds softer, more spontaneous, more about a passing craving
• я хочу can sound more direct or firm

In this sentence, мне иногда хочется fits very naturally because the speaker means:

• sometimes after dinner I feel like having exactly that

not necessarily a strong deliberate decision.

Because его is the object form, while он is the subject form.

Here the speaker is saying they want it/him, not that it/he is doing something.

• он = he / it as subject
• его = him / it as object (and also genitive form)

So:

• Он маленький. — It is small.
• Мне хочется именно его. — I want exactly that one.

Since мандарин is an inanimate masculine noun, English uses it, but Russian still uses masculine grammatical forms.

Also, after хочется, Russian often uses an object form that corresponds to genitive or accusative depending on context. With его, the form is the same either way, so you do not see a difference here.

Именно adds emphasis. It means something like:

• exactly
• precisely
• that very one

So:

• мне хочется его = I want it
• мне хочется именно его = I want that one specifically / I want exactly that

In this sentence, именно helps set up the contrast with:

• а не большой десерт
  and not a big dessert

So the speaker means: even though the mandarin is small, that is specifically what they feel like after dinner.

Because а is often used for contrast or correction, especially in patterns like:

• X, а не Y = X, not Y

Here the speaker is contrasting two possibilities:

• именно его
• а не большой десерт

So the meaning is:

• exactly that, rather than a big dessert
• that and not a big dessert

Using но would sound less natural here. Но is more like but, while а is very common when setting one thing against another.

Compare:

• Я хочу чай, а не кофе.
  I want tea, not coffee.

• Это маленький фрукт, а не десерт.
  This is a small fruit, not a dessert.

Because десерт is functioning as the contrasted object, and for an inanimate masculine noun in the accusative singular, the form is the same as the nominative:

• nominative: большой десерт
• accusative (inanimate): большой десерт

So although it looks like nominative, here it is understood as the object in the contrast:

• мне хочется именно его, а не большой десерт

If the noun were animate, you would often see a different form in the accusative. But with inanimate masculine nouns, nominative and accusative match.

Because the first part is a separate concessive clause:

• Пусть мандарин и маленький, ...

Then the main clause follows:

• после ужина мне иногда хочется именно его, а не большой десерт.

So the comma separates:

1. the although/even if part
2. the main statement

This is very normal punctuation for this kind of sentence.

Yes, in this context it clearly means a mandarin orange / tangerine.

Russian мандарин can mean:

• the fruit mandarin
• historically, a mandarin (official in imperial China)

But with:

• маленький
• после ужина
• хочется именно его
• а не большой десерт

the meaning is obviously the fruit.

Yes, it is understandable and fairly natural. The overall idea sounds like:

• Although a mandarin is small, after dinner I sometimes feel like exactly that, not a big dessert.

A native speaker might also phrase it in slightly different ways, for example:

• Пусть мандарин и маленький, но после ужина мне иногда хочется именно его, а не большого десерта.
• Хотя мандарин и маленький, после ужина мне иногда хочется именно его, а не большого десерта.

But the original sentence works, especially if the goal is to emphasize the contrast between a small fruit and a large dessert.