Definitely claim your mileage if you work different locations on an irregular basis. If you’re working one location on a regular day of the week then you should be PAYE for that employer.
I don’t think the basic tests of self employment have changed for a long time; check out the AOP advice for more detail but in summary;
You bring your own equipment.
You are free to send a substitute to complete the contracted work.
You have no contract (written or verbal) to attend on a regular basis (this doesn’t affect block cover for holidays or maternity as they have a definite duration).
You have no employee related benefits (holiday, sick, xmas meal etc.)
Some others I think.