Check local hospitals for a nurse-supervised sick unit, check Care.com for a bonded babysitter - not cheap but you need a back-up plan. Check colleges for students who will serve as part-time nannies.
My stepdaughter lost a job for the very reason you state - companies may be sympathetic, but they cannot operate with an unreliable work force.
I'm sure you wish you had a plan in place from the moment you got the job, but don't beat yourself up about it now. Move forward. You need a sick plan, a snow day plan, a teacher in-service day plan, a school power failure plan, and a school vacation day/holiday plan. You also need a "Super Short Notice" plan for when you wake up and she's throwing up at 7:30 AM. At least now you have a "night before" warning. Not a lot of time, but better than the day of. When I was teaching, we often had kids coming to school and throwing up at 8:30, when you just know that kid was nauseated at home at 7:30. Parents go to work, send a sick-ish kid to school, and get a call at work an hour later. That's not appreciated by employers either.
As a single parent, you have extra pressures, so you need more plans.
If you can find a neighbor/friend who has a child at home, you might consider trading off (give her some Saturday nights out with her husband if you have to) - but sometimes people don't want a sick kid in with their well kid.
Check with your town's senior center to see if there are able-bodied retirees looking for a few bucks now and then. Maybe you can put up a flyer and interview a couple of people in advance. There are home health companies that provided assistance for people (often elderly, sometimes post-surgery) and they do respond on short notice, but they can charge upwards of $23 an hour, and that's going to break your budget if you do it a lot.
Good luck!