If someone looked after foster children on weekends (from Friday afternoon until Sunday afternoon) and received remuneration does this count as work and not permissible on the Sabbath?
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As Ariel stated, in theory one can perform services (not physically-creative actions like plowing, planting, cooking, writing, burning and the like) such as childcare, waiting/bussing tables, teaching/sermonizing/performing prayer services if there is a lump payment for a period of time that includes some non-Shabbat.
Thus you could pay a babysitter for "the period 6-8PM", where 6-7PM is shabbat and 7-8PM isn't; or a rabbi for "answering questions during the week and delivering a sermon on shabbat."
In theory I don't see why foster care should be different, but as always ask your rabbi.