Learn Hardcore German: Meine Mutter hofft auf eine kleine Gehaltserhöhung, weil die Stromrechnung immer höher wird. - My mother is hoping for a small pay raise because the electricity bill keeps getting higher.

Main clause: Die Stromrechnung wird immer höher. (verb in 2nd position)
With weil: ..., weil die Stromrechnung immer höher wird. (verb at the end)

Saying ..., weil die Stromrechnung wird immer höher is incorrect in standard German.

What does hofft auf mean, and why do we need auf here?

The verb is auf etwas hoffen = to hope for something.
In German, you normally cannot just say hoffen with a direct object; you need the preposition auf with the accusative:

Meine Mutter hofft auf eine kleine Gehaltserhöhung.
Literally: My mother hopes for a small salary increase.

You could also say Meine Mutter hofft, dass sie eine kleine Gehaltserhöhung bekommt, but then the structure changes completely.

Why is it eine kleine Gehaltserhöhung and not einen kleinen Gehaltserhöhung?

Gehaltserhöhung is a feminine noun (die Gehaltserhöhung).

In auf eine kleine Gehaltserhöhung:

The preposition auf in this meaning (to hope for something) takes the accusative.
Feminine accusative singular with an indefinite article is eine.
The adjective before a feminine accusative singular with eine gets the ending -e → kleine.

So:

Nominative: eine kleine Gehaltserhöhung
Accusative: eine kleine Gehaltserhöhung

(They look the same in the feminine.)

How is Gehaltserhöhung formed, and what does it literally mean?

Gehaltserhöhung is a compound noun:

das Gehalt = salary
die Erhöhung = increase

Together: die Gehaltserhöhung = an increase of salary, i.e. a pay rise / raise.
German very often creates long nouns by combining two (or more) shorter ones like this.

Why is it Meine Mutter and not Meine Muter or Mein Mutter?

Two separate points:

Spelling: Mutter is always written with double t in German: die Mutter.
The possessive: Mutter is feminine. In the nominative singular, the correct possessive form is meine:

meine Mutter (my mother)
mein Vater (my father, masculine)

Because Mutter is the subject of the sentence, it is in the nominative: Meine Mutter hofft ...

Why is Stromrechnung written as one word? Could it be two words?

In German, nouns are often combined into one long compound noun. Stromrechnung is:

der Strom = electricity (as a utility)
die Rechnung = bill, invoice

Together: die Stromrechnung = electricity bill.

Spelling it as two separate nouns (Strom Rechnung) would be incorrect in standard German. It must be written as one word.

What case is die Stromrechnung, and how do we know?

In the clause weil die Stromrechnung immer höher wird, die Stromrechnung is the subject of the verb wird.
Subjects are in the nominative case.

Clues:

It answers who or what is becoming higher? → die Stromrechnung.
The article is die, which is nominative feminine singular for Stromrechnung.

Why is it immer höher and not something like mehr höher?

Höher is already the comparative form of hoch (high → higher).

In German, you normally do not add mehr in front of a regular comparative adjective; you don’t say mehr höher, just like English does not say more higher.
Immer höher literally means ever higher / higher and higher / more and more expensive in this context.

So:

hoch = high
höher = higher
immer höher = higher and higher / more and more high

Why does hoch become höher and not hocher?

The adjective hoch has a slightly irregular comparative:

hoch → höher

Two things happen:

The vowel gets an umlaut: o → ö.
The -ch drops before the comparative ending -er.

This is just one of a small group of irregular comparatives in German, and it’s something you simply have to memorize.

What exactly does immer add to the meaning in immer höher wird?

Immer here means more and more or increasingly.

höher wird = is becoming higher
immer höher wird = is becoming higher and higher / is continuously increasing

So it emphasizes the ongoing, repeated increase, not just a one-time change.

Why is wird used instead of ist in immer höher wird?

Wird is from werden and expresses change or becoming:

Die Stromrechnung ist hoch. = The electricity bill is high. (a state)
Die Stromrechnung wird höher. = The electricity bill is getting higher. (a change)

With immer höher wird, the focus is on the bill continuously increasing, not just being at a high level.

Could we use denn instead of weil here, and what would change?

Yes, you can say:

Meine Mutter hofft auf eine kleine Gehaltserhöhung, denn die Stromrechnung wird immer höher.

The meaning (because) stays almost the same, but:

Weil is a subordinating conjunction → verb goes to the end: ..., weil die Stromrechnung immer höher wird.
Denn is a coordinating conjunction → normal main-clause order (verb in 2nd position): ..., denn die Stromrechnung wird immer höher.

Stylistically, weil sounds slightly more colloquial and is more common in spoken German; denn is often a bit more formal or written.

Why do we need a comma before weil?

In German, every full subordinate clause is separated from the main clause by a comma.
Since weil die Stromrechnung immer höher wird is a subordinate clause introduced by weil, you must put a comma before it:

Meine Mutter hofft auf eine kleine Gehaltserhöhung, weil die Stromrechnung immer höher wird.

Leaving the comma out here is considered a spelling mistake in standard written German.

Why is kleine lowercase, but Gehaltserhöhung and Stromrechnung are capitalized?

In German, all nouns are capitalized:

die Gehaltserhöhung
die Stromrechnung
die Mutter

Adjectives like kleine are not capitalized in normal use, so:

eine kleine Gehaltserhöhung (adjective + noun)

The capitalization signals that Gehaltserhöhung and Stromrechnung are nouns.

Could we leave out kleine and just say eine Gehaltserhöhung? Would the grammar change?

Yes, you can say:

Meine Mutter hofft auf eine Gehaltserhöhung, weil die Stromrechnung immer höher wird.

The grammar and structure stay the same; you just remove the adjective.
The meaning changes slightly: eine Gehaltserhöhung is neutral about size; eine kleine Gehaltserhöhung suggests a modest or small raise.

Why is auf followed by the accusative here and not the dative?

The preposition auf can take either accusative or dative, depending on meaning:

auf
- accusative: usually direction / movement toward something (Wohin?)
auf
- dative: usually position / location (Wo?)

However, some verb–preposition combinations are fixed and always use one case.
Auf etwas hoffen is an idiomatic verb–preposition pair that always uses the accusative, even though no physical movement is involved:

auf eine Gehaltserhöhung hoffen (accusative)

So eine kleine Gehaltserhöhung must be in the accusative.