I'm new on here and not sure how to edit my post.
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Your question is partly to do with Employment Status. This as you may know from browsing questions asked on here is a tricky subject.
Your starting point is to use the HMRC Employment Status Indicator to get a view.
You say you are "helping out a third family with 2 morning school drop offs (7-9am) and 2 hours of ironing per week term time only." So does that mean you have two other employed jobs already, which are under contract and you are getting payslips showing tax deductions?
Two mornings a week, same days each week, term time only, is still a job - the parents are most likely expecting commitment from you to turn up and to do the work - work they determine, they can tell you what to do, moan when you don't do it to their standard. They have a lot of control even if they may not actually exercise that control all the time.
If this was your only job, then the pay would be below the lower earnings limit (£112 per week) and thus the parents would not need to do anything. As it is not your only job and as it is regular work not ad hoc, I feel it is likely to be classed as employment. That then gives you employment rights - such as paid holiday (which they could insist is taken during school holidays) and right to being given notice if the job is to end. More about Rights At Work
6 hours x 38 weeks = 228 hours.
Holiday at 12.07% = 27.52 hours.
Total Hours Per Year: 255.52 hours.
It this was paid at £12 gross an hour, it would be £3,066.24 perhaps rounded up to 256 hours, £3072. Short hours can be paid reasonably highly as it is not easy to find someone to do that work. You will need to look at local area job rates to get a feel for the likely salary in your specific area.
In 2016/17 tax year (which starts in April), with a BR tax code (as you have other work) there would be a deduction of £614.40 for income tax. There is no employee NI, no employers NI. So the parents cost would be £3072 + Payroll Admin (around £180 a year if you are paid monthly - Parents Guide To Employing A Professional Nanny).
As an employee you get mileage payments for using your own car for work.
This can be up to 45p per mile before it becomes taxable. For a small car, it may work out around 30/35p a mile - depending on the miles done. The lower the mileage, the higher the per mile cost. You would be paid for the journey made from their home to the school and back to their home. If you took the children to school then did not head back to their home, you could negotiate to be paid the return trip as if it was really made, as long as the real journey from the school to your home is longer. Your home may be many miles away from the school, or it may be close to the school. You should not be out of pocket for transporting the children but neither should you make a profit, so you need to find the balance point.
One day per week would you not be going back to their home to do the ironing?