Yes there are significant and multiple areas where private healthcare companies are being paid to 'provide' services to the NHS in a classic outsourcing process similar to what you would see in IT.

Private company charge a lot more, pay their staff a bit more to attract workers from the NHS, extend their contracts to provide more services where there are NHS staff shortages. Taxpayer money turns into company profits. I wouldn't object if it provided more efficient care but it doesn't seem to.

It's a bit complicated. Private providers can have different staff mixes, so you might have more healthcare assistants and fewer nurses. You might have more unregistered nurses and fewer registered nurses. You might find more staff are on zero-hour contracts, or things like "hotel services" (food, cleaning) are contracted out to companies that are low pay poor condition companies.

Don't most countries in the EU have a mix of public and private health care providers? How do they manage it?

In England we can talk about NHS funding and private funding, and we have both.

We can also talk about NHS providers and private providers of NHS services, and again we have both.