Trzymam kubek mocno, żeby nie rozlać herbaty.

Trzymam is the 1st person singular present tense of trzymać (to hold, to keep/keep holding).

So trzymam kubek means I am holding a mug or I hold the mug.

A useful thing to remember: Polish present tense often covers both English simple present and present continuous. So trzymam can mean I hold or I am holding, depending on context.

Because Polish usually drops subject pronouns when the verb already makes the subject clear.

The ending -am in trzymam already tells you the subject is I, so ja is unnecessary.

You could say Ja trzymam kubek mocno..., but that would usually add emphasis, for example:

• Ja trzymam kubek, nie ty. — I’m holding the mug, not you.

Because kubek is the direct object of trzymam, so it is in the accusative case.

However, kubek is a masculine inanimate noun, and for many masculine inanimate nouns, the accusative singular looks the same as the nominative singular.

So:

• nominative: kubek
• accusative: kubek

That is why you do not see a change here.

Mocno is an adverb, and here it means something like firmly, tightly, or securely.

You use an adverb because it describes how the action is done:

• trzymam mocno — I hold firmly

By contrast, mocny is an adjective, used to describe a noun:

• mocny chwyt — a strong grip
• mocny kubek — a sturdy mug

So in this sentence, mocno is correct because it describes the verb trzymam.

Żeby introduces a purpose clause. Here it means so that, in order to, or so as not to.

So:

• Trzymam kubek mocno, żeby nie rozlać herbaty. means:
• I’m holding the mug firmly so that I don’t spill the tea.

This is a very common Polish structure:

• robię coś, żeby... — I do something so that...
• zamknąłem okno, żeby nie było zimno — I closed the window so that it wouldn’t be cold

Because in Polish, when the subject of both parts is the same, it is very natural to use żeby + infinitive.

Here, the person who is holding the mug is also the person who might spill the tea, so żeby nie rozlać works well.

A more explicit version is also possible:

• Trzymam kubek mocno, żebym nie rozlał herbaty.
  or, if the speaker is female:
• ... żebym nie rozlała herbaty.

That version sounds a bit more explicit because it marks the subject and even the speaker’s gender. The infinitive version is shorter and very natural.

This is about aspect.

• rozlać is perfective
• rozlewać is imperfective

Here, rozlać is used because the speaker wants to avoid one complete accidental result: spilling the tea.

So:

• nie rozlać herbaty = not to spill the tea
• nie rozlewać herbaty would sound more like not to be spilling tea, not to spill it repeatedly, or it would focus on the process rather than the single result

That is why rozlać is the natural choice here.

Because the infinitive rozlać is negated by nie, and in standard Polish, a negated direct object often goes into the genitive instead of the accusative.

So with herbata:

• accusative singular: herbatę
• genitive singular: herbaty

Compare:

• Rozlałem herbatę. — I spilled the tea.
• Nie chcę rozlać herbaty. — I don’t want to spill the tea.

In your sentence, the same pattern appears:

• żeby nie rozlać herbaty

This is a very important case pattern in Polish.

In Polish, nie normally goes before the verb or infinitive it negates, and with most verbs it is written separately.

So:

• nie rozlać — not to spill
• nie trzymam — I am not holding

That is why you see:

• żeby nie rozlać herbaty

There are some spelling rules elsewhere in Polish where nie is written together, but with verbs it is generally written separately.

Polish word order is fairly flexible, but different orders change the emphasis.

The original sentence is neutral and natural:

• Trzymam kubek mocno, żeby nie rozlać herbaty.

Other possible versions include:

• Mocno trzymam kubek, żeby nie rozlać herbaty.
• Kubek trzymam mocno, żeby nie rozlać herbaty.

These can sound more marked or emphasize a different part of the sentence.

The most important thing is that the purpose phrase żeby nie rozlać herbaty stays clear and attached to the main idea.