"Don't mention child care! That's inappropriate and makes it sound like a hobby business!"
God forbid that a woman should actually admit she has children 😱. Professional women should act at all times like their children do not exist.
In the case of the OP she missed her children's match because of work. Hence she must have had some form of childcare, whether formal and informal. If she was paying for childcare, the missed lessons would hurt even more as she would bear the cost of childcare without being paid. Of course you can point that out to clients. Believe it or not, most clients are aware of your children as private lessons may take place in your home. As parents themselves, they understand the juggling act which goes on. Tuition takes place after school and at the weekend which means there are frequent clashes with activities, clubs and so on.
"Don't "hope" or expect charity "fingers crossed" for some moral trigger."
Of course you can hope and cross your fingers that your clients will honour your agreement. Not to expect charity, just to hope that they don't make you chase them for the money, which can be a massive waste of your time and energy. At the end of the day, unless they pay willingly, your only form of comeback would be to pursue through small claims, and the amount/costs involved would mean this is not worth doing. Even sending a solicitor's letter would probably be the same amount as you were trying to recover. You are more likely to fire them as a client and write it off as a bad debt. Most tutors, of course, mitigate this risk by asking for the money up front monthly or pay on the day.
"Would we advise a male PT, tutor, etc to mention childcare and "hope" for payment? No? Because it's stupid unprofessional advice."
Meanwhile, in the real world, a few weeks ago my male mechanic asked me to pick my car up early so he could leave to pick his kids up from school. Last week my husband re-arranged some work as he was due to look after the children that day. It's not stupid or unprofessional to mention your children - it's reality. Tutors are people and parents too 🤷♀️