Should there be an Indefinite article in a quote "Looking for work" or that is a correct way to say it?
As I see it, it's just "the way we say it in English" is that right?
This sense of 'work' is not countable.
Work (MW, noun definition 1a)
activity that a person engages in regularly to earn a livelihood - people looking for work
Contrast that with 'job', which is countable.
Now that I've graduated from college, I need to get a job.
She has 2 jobs; she's a full-time secretary, and she tends bar on Friday and Saturday nights.