Not exactly. Work is a more abstract way of defining a job. You cannot have "a work", but you can have " a job".
'An' signifies 'one.' In Middle English (think around the 1300s-1400s), 'an' meant 'one' or 'single' or 'alone.' The two words - an other - naturally join together in speech because of the elision between n and o, so eventually they were joined in writing as well.
Aince 'an' means 'one;' another means'' 'one other.' It would not make sense to repeat the 'an' (one one other is senseless).
'Other' would take a plural: do you have other jobs. Another means 'one other.' Do you have ONE other job - thus the singular. Since the noun in the Polish sentence is singular, that tells you that you need to use either 'another' or 'a different' in your answer.
In Polish, 'inne' means 'other' or 'another' or 'different' depending on the context and the number of the verb that follows.
I didn't try, 'Do you have other work,' but grammatically, that would work in English, since 'work' is uncountable ('job' is countable: a/one job, three jobs; but: some work; different work).
The audio is not playing for me, making the "write what you hear" version of this impossible to get right without failing once.
Listening questions have a button labelled "can't listen now". It's for situations exactly like this.
True, the audio file doesn't seem to exist for the male voice. I reported it to the devs (but I wouldn't count on them fixing it soon) and disabled the audio exercises so no one gets the 'write what you hear' exercise again.