После долгой прогулки всем захотелось добавки, особенно сыну.

Because после always takes the genitive case.

• прогулка is the dictionary form (nominative singular).
• After после, it becomes прогулки (genitive singular).
• The adjective has to agree with the noun in gender, number, and case, so долгая becomes долгой.

So:

• долгая прогулка = a long walk
• после долгой прогулки = after a long walk

Because the verb pattern here is кому-то захотелось чего-то.

With хотеться / захотеться, the person who feels the desire is usually put in the dative, not the nominative.

Examples:

• мне захотелось спать = I felt like sleeping
• детям захотелось мороженого = the children wanted some ice cream
• всем захотелось добавки = everyone wanted more / a second helping

So всем literally means to everyone.

This is an impersonal construction.

There is no normal grammatical subject in the nominative case, so Russian uses the default past-tense form neuter singular: захотелось.

That is very common with verbs like:

• хотеться
• нравиться
• нездоровиться
• не спится

So:

• все захотели добавку = everyone decided/wanted more (personal construction)
• всем захотелось добавки = everyone felt like having more (impersonal, very natural)

The second version often sounds softer and more idiomatic.

The difference is mainly aspect.

• хотелось = an ongoing state, felt like
• захотелось = the desire arose, suddenly felt like

In this sentence, захотелось fits well because it suggests that after the long walk, the desire appeared.

Compare:

• После прогулки всем хотелось есть. = After the walk, everyone was hungry / felt like eating.
• После прогулки всем захотелось добавки. = After the walk, everyone started wanting more.

So захотелось gives a clearer sense of a new desire appearing at that moment.

Because after хотеться / захотеться, the thing wanted is often put in the genitive, especially when it means some amount of something rather than one specific item.

Here добавки is the genitive singular of добавка.

This use is very natural when talking about food:

• хотелось супа = felt like some soup
• просить добавки = ask for more / ask for a second helping
• захотелось добавки = wanted more / wanted a second helping

So the genitive here has an indefinite quantity feel: not one clearly defined object, but some more.

Here добавка means an extra portion, more food, or a second helping.

It is commonly used in meal contexts:

• Дать добавки? = Shall I give you some more?
• Он попросил добавки. = He asked for a second helping.

So in this sentence, it does not just mean addition in an abstract sense. It specifically means more of the food.

Because сыну is still part of the same grammatical pattern as всем.

The sentence says:

• всем захотелось = everyone felt like it
• особенно сыну = especially the son

Since the experiencer of захотелось is in the dative, сын also has to be in the dative: сыну.

You can think of it as:

• всем, особенно сыну, захотелось добавки

So особенно сыну means especially the son within that same dative structure.

Russian word order is flexible, and the end of the sentence often carries special emphasis.

Putting особенно сыну at the end highlights it:

• После долгой прогулки всем захотелось добавки, особенно сыну.

This feels natural because the sentence first gives the general idea, then adds the pointed detail: especially the son.

A different order is possible, but the emphasis changes. For example:

• Особенно сыну после долгой прогулки захотелось добавки.

That puts stronger early focus on the son, but it sounds less neutral.

Yes, but it is not exactly the same in tone.

• все захотели добавку = everyone wanted an extra portion
• всем захотелось добавки = everyone felt like having more

The version with захотелось is more idiomatic for a spontaneous inner desire. It sounds a bit less direct and often more natural in this kind of context.

Also, добавки in the genitive works especially well with the idea of some more food, while добавку sounds more like a specific portion.

So both are possible, but the original sentence has a very natural Russian feel.