Więc means so, therefore, or thus.
In this sentence, it connects cause and result:
So the whole structure is basically:
It is a very common conjunction in everyday Polish.
Móc is the infinitive, meaning to be able / can.
In the sentence, you need the conjugated form because the subject is I (understood in Polish):
So:
This is just the 1st person singular present tense form of móc.
Polish often drops subject pronouns when the verb ending already shows who the subject is.
Here, mogę clearly means I can, so ja is unnecessary.
Compare:
Both are possible, but leaving out ja is more natural unless you want emphasis.
Się is a reflexive particle that is part of many Polish verbs. In this case, zalogować się means to log in.
You should learn it as a whole expression:
English does not use a reflexive word here, but Polish does. So się is not optional.
This is about aspect, which is very important in Polish verbs.
Very roughly:
After móc / nie móc, Polish often uses the infinitive that fits the intended meaning. Here, nie mogę się zalogować means:
That is why the perfective zalogować się is very natural here.
In Polish, się often appears before or after the verb, but not usually at the very beginning of a clause.
So these are common patterns:
This placement is natural and standard. You may also sometimes see zalogować się when the infinitive stays together as a dictionary form, but in full sentences się often shifts position.
Here nie mogę is followed by an infinitive verb, not a noun, so there is no special noun case issue in this part.
Structure:
This is very common in Polish:
So the sentence is built with a modal verb (móc) plus an infinitive.
In many computer-related contexts, zalogować się covers log in, sign in, and sometimes log on.
So this sentence is very natural for all of those meanings:
Depending on context, Polish speakers may also use similar expressions like:
But zalogować się is the standard verb for logging in.
It could be. The sentence as given is simply shorter because the context is already clear.
You can say:
Or more explicitly:
The shorter version avoids repetition, since system was already mentioned.
For machines, software, and systems, działać is the normal verb for to work / to function.
Pracować usually means to work in the sense of a person working, or a machine operating in some contexts, but for software/system failure działać is much more natural.
So System nie pracuje would sound odd in most everyday IT contexts.
A rough pronunciation guide is:
A few helpful notes:
Polish stress in this whole sentence is regular and predictable in most words.