Po deszczu na ścieżce jest błoto, a rano nad rzeką była mgła.

Because po in the sense of after takes the locative case in Polish.

• base form: deszcz = rain
• after po: po deszczu = after the rain

A useful detail: the form deszczu is the same in both the genitive and locative singular, but here it is locative because it follows po.

Because na can take different cases depending on meaning:

• na + locative = location, on / at
• na + accusative = movement toward a place, onto / to

So:

• na ścieżce = on the path / on the trail (location)
• na ścieżkę = onto the path (movement)

In this sentence, the mud is already there, so Polish uses the locative: na ścieżce.

It is the locative singular form of ścieżka.

• nominative: ścieżka
• locative: ścieżce

This change is normal for many feminine nouns ending in -ka. The k often changes to c before the ending:

• ścieżka → ścieżce
• compare also: książka → książce

Because nad changes case depending on whether you mean location or movement.

• nad + instrumental = location, over / above / by
• nad + accusative = movement, to over / toward

So:

• nad rzeką = over the river / by the river (location)
• nad rzekę = to the river / over the river (movement toward that area)

Here the fog was located there, so Polish uses the instrumental: nad rzeką.

Because Polish past tense agrees with the gender of the subject, but the present tense singular does not show gender in the same way.

• mgła is feminine singular, so in the past: była mgła
• błoto is neuter singular, so if it were in the past, it would be było błoto

But in the present singular, all genders use jest:

• jest mgła
• jest błoto

So:

• jest błoto = there is mud
• była mgła = there was fog

Because Polish does not need a dummy subject like English there.

English says:

• There is mud on the path.

Polish simply says:

• Na ścieżce jest błoto.
• literally: On the path is mud.

That is a normal Polish way to express existence. The idea of there is / there was is usually just conveyed by jest / był(a/o) plus the noun.

In this sentence, a links two clauses, but it often has a slight sense of contrast, scene shift, or meanwhile / and.

• i = straightforward and
• a = and, but often with a change of topic, setting, or comparison

Here the sentence moves from:

• after the rain, on the path to
• in the morning, over the river

So a sounds very natural. It is not a strong but, but it is not quite the same as plain i either.

Rano is an adverb meaning in the morning or sometimes this morning, depending on context.

Polish often uses bare time adverbs without a preposition:

• rano = in the morning
• wieczorem = in the evening
• dzisiaj = today
• jutro = tomorrow

So rano nad rzeką była mgła is perfectly normal.
You do not say w rano.

Because the sentence describes two different time frames:

• jest błoto = there is mud now / at the relevant present moment
• była mgła = there was fog earlier, in the morning

That mix is completely normal if the speaker means:

• now, after the rain, the path is muddy
• earlier this morning, there was fog over the river

If you wanted both parts fully in the past, you could say:

• Po deszczu na ścieżce było błoto, a rano nad rzeką była mgła.

No, Polish word order is fairly flexible. The order here is natural, but other orders are possible if you want to emphasize something different.

For example:

• Po deszczu na ścieżce jest błoto
• Na ścieżce jest błoto po deszczu
• Rano nad rzeką była mgła
• Nad rzeką rano była mgła

The basic meaning stays similar, but the focus changes. Polish often puts known context first and new information later.

Because Polish has no articles.

English distinguishes:

• a path
• the path

Polish simply says:

• ścieżka / ścieżkę / ścieżce, etc.

So na ścieżce can mean on the path or on a path, depending on context. The same applies to nad rzeką.

Because in Polish, when a joins two independent clauses, a comma is normally used.

So:

• Po deszczu na ścieżce jest błoto, a rano nad rzeką była mgła.

This is standard Polish punctuation. English punctuation is often a bit looser here, but in Polish the comma before a is expected.

Yes:

• po deszczu

  • deszczu = locative singular after po

• na ścieżce

  • ścieżce = locative singular after na for location

• nad rzeką

  • rzeką = instrumental singular after nad for location

And the subjects are:

• błoto = nominative singular
• mgła = nominative singular

So this one sentence is a good example of how Polish prepositions control different cases depending on meaning.