Do people work to support their families intentionally or incidentally?
The question here is whether the work performed to support a family (or simply one's self) is intentional: that is, done for its own sake as well as for the ability to pay bills (done deliberately, with intent) -- or incidental: that is, performed only as a necessity to afford the ability to live (a minor accompaniment to something larger).
Supporting a family is an important responsibility; the breadwinner(s) is/are responsible for keeping the family alive, fed, housed, and so forth. To this end, work is the means by which the family is supported; the worker goes to a job, works, and is able to keep the family alive until the next generation is able to work and contribute. In this model, the work itself must be intentional (deliberately done), because a bad worker will be fired; the job may come from intentional, incidental, or accidental events, but working to support the family requires a certain amount of intentional ability and perseverance.
If the work is incidental (minor accompaniment), the worker is tasked with enduring a potentially-hazerdous, difficult, or simply boring job because of the need to support the family. In this case, the work may be entirely arbitrary; the worker might not enjoy the job or even think about it outside of work, and only worry about performing the bare minimum to ensure a monthly paycheck.
Intentional work is usually done because the worker "found their calling," or discovered an innate talent that serves the labor. Incidental work may often be piecemeal or paycheck-to-paycheck, including many minimum-wage jobs; the worker must work or the family will starve. In either case, the work is the means by which the family is supported; intentional work is done for a higher purpose, and so the end result is that most workers work can be classified as incidental more than intentional.