In the case of a bar or a restaurant, you could say "pick up the tab/check".

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You don't "clear the bill". In a place where a bill exists, you pay it. That doesn't happen at a supermarket though. It happens at a bar or a restaurant, or any place where you could be presented with a statement of expenses you have to settle.
You could say I pay the cashier at the check-out at the supermarket.
I am not a teacher.
In the case of a bar or a restaurant, you could say "pick up the tab/check".
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
That's right. I use it if I pay for other people.