Overheating

The water pump sits in the coolant. You do not add anything to it. There are a lot of reasons for overheating but if the coolant level goes down, that suggests a cylinder head gasket is leaking. White smoke from the tail pipe is one clue. Your mechanic can also do a chemical test at the radiator or reservoir to verify that. A properly running engine with no leaking head gasket can not overheat in just ten minutes. If air leaks into the cooling system from a leaking head gasket, it can pool under the thermostat. That air will prevent the thermostat from opening since they open in response to hot liquid, not hot air. The clue would be when you see the temperature gauge on the dash go up to "hot", the radiator hoses will still be cold. That could cause it to overheat that quickly.

There could also be an air pocket simply because the coolant level went low over a long period of time. Once you filled the system that air may need to be bled out so the coolant will go up to the thermostat. I do not know which engine you have so you will have to look if there is a bleeder screw on the thermostat housing or if there is a threaded plug or sensor on or near that housing that can be unscrewed to bleed the air out.

I would say start by changing the thermostat they are cheap. Then drain you coolant see if it is dirty or needs to be flushed could possibly be a clogged radiator. If that does not fix it have a compression test done it could be a blown head gasket.