После прогулки мне захотелось простой горячей картошки.

Because после always takes the genitive case.

• прогулка = nominative
• после прогулки = after the walk / after a walk

So this is just a case change required by the preposition после.

A useful pattern:

• после урока = after the lesson
• после работы = after work
• после прогулки = after the walk / after a walk

Russian also has no articles, so после прогулки can mean either after the walk or after a walk, depending on context.

This is a very common Russian way to talk about a feeling or desire that comes over someone.

• мне is dative
• захотелось is an impersonal verb form

So the structure is roughly:

• мне захотелось X = I felt like X / I suddenly wanted X

Literally, it is closer to to me, it became desirable than to I wanted.

This construction often sounds more natural when talking about cravings, moods, or spontaneous wishes:

• Мне захотелось спать. = I suddenly felt sleepy / I felt like sleeping.
• Мне захотелось чаю. = I felt like some tea.

Захотелось is:

• the past tense
• of the perfective verb захотеться
• used impersonally
• with the reflexive ending -ся

A few important points:

1. Perfective
   It shows that the desire arose at a particular moment.

2. Past tense neuter singular
   In impersonal past-tense constructions, Russian often uses the neuter singular form by default:

   • захотелось
   • стало
   • получилось

3. Impersonal meaning
   There is no normal grammatical subject like я here.

So мне захотелось means something like:

• I got the urge
• I suddenly felt like
• I found myself wanting

These forms are related, but the nuance changes.

• мне захотелось = I suddenly felt like / the desire came up
• мне хотелось = I wanted / I felt like, more ongoing or background
• я хотел = I wanted, with a more direct personal subject

Compare:

• После прогулки мне захотелось картошки.
  After the walk, I suddenly felt like some potatoes.

• После прогулки мне хотелось картошки.
  After the walk, I was wanting some potatoes / I felt like some potatoes.

• После прогулки я хотел картошку.
  After the walk, I wanted potatoes.

The sentence with мне захотелось feels especially natural for a spontaneous craving.

Here картошки is in the genitive, and that often happens when Russian talks about an indefinite amount of food or drink.

So:

• картошки here means something like some potatoes / some potato
• not one clearly defined, specific object

This is very common after verbs of wanting, craving, eating, drinking, etc.:

• хочу чаю = I want some tea
• мне захотелось супа = I felt like some soup
• поесть хлеба = to eat some bread

If you used картошку instead, it would sound more like a more specific, more concrete object or dish. That version is possible in some contexts, but картошки is very natural here because the idea is a craving for some hot potato(es).

Because adjectives must agree with the noun they describe in:

• gender
• number
• case

The noun here is картошки, which is:

• feminine
• singular
• genitive

So the adjectives must also be feminine singular genitive:

• простая → простой
• горячая → горячей

Compare:

• простая горячая картошка = simple hot potato(es) / simple hot potato dish
  nominative

• простой горячей картошки = of some simple hot potato(es)
  genitive

In this sentence, картошка is being used as a food word, not as a count of individual potatoes.

So it can mean something like:

• potatoes
• potato
• a potato dish

depending on the situation.

That is why English often has to choose a natural translation such as:

• some hot potatoes
• some plain hot potato
• a simple hot potato dish

Russian картошка is very flexible in everyday speech when talking about food.

Картошка is the everyday, conversational word.
Картофель is more formal, neutral, or technical.

Compare the tone:

• картошка = casual, homely, everyday
• картофель = more formal, menu-like, agricultural, or official

Since this sentence is about a personal craving after a walk, картошки sounds warm and natural. Картофеля would sound less cozy and more formal.

No. Here простая means:

• simple
• plain
• not fancy
• basic, uncomplicated

So простая горячая картошка suggests something modest and satisfying, not an elaborate dish.

It does not mean easy potato.
It means something more like:

• plain hot potatoes
• simple hot potato
• just some basic hot potatoes

Because Russian often leaves that idea unstated when it is obvious from context.

If someone says:

• мне захотелось картошки

the natural understanding is:

• I felt like eating some potatoes

Russian does not need to spell out есть every time.

If you wanted to make it explicit, you could say something like:

• Мне захотелось поесть картошки. = I felt like eating some potatoes.

But the shorter version is very natural and common.

The basic meaning stays the same, but the emphasis can change.

Original:

• После прогулки мне захотелось простой горячей картошки.

This puts after the walk first, setting the scene.

Other possible orders:

• Мне после прогулки захотелось простой горячей картошки.
• Простой горячей картошки мне захотелось после прогулки.

All of these are possible, but they highlight different parts of the sentence.

Russian word order is more flexible than English, especially because case endings show grammatical relationships.

Yes, you could, but it feels a bit different.

• я захотел sounds more direct and personal: I wanted
• мне захотелось sounds more like a spontaneous feeling or craving: I suddenly felt like

In a sentence about food cravings, мне захотелось is often the more natural choice.

So:

• После прогулки мне захотелось простой горячей картошки.
  very natural for a sudden craving

• После прогулки я захотел простой горячей картошки.
  possible, but a bit more deliberate or narrative

The order is not as fixed as in English.

Here:

• простой = simple/plain
• горячей = hot

So простой горячей картошки means some plain hot potatoes.

Russian adjective order is somewhat flexible, but some orders sound more natural than others. In this sentence, простой горячей картошки sounds smooth and natural.

If you reversed them to горячей простой картошки, it would still be understandable, but it would sound less natural in most contexts.