Ok so pictures weren’t the best, I don’t know which gray cap you are referring to but I will try to answer your question.

1.) If the radiator was replaced and it’s still leaking check the radiator cap, they do fail after a while it’s basically a spring seal idea, the seals and all will fail eventually.
2.) While the car is warm (NOT RUNNING) actually squeeze both lower and upper radiator hoses sometimes you can have pin holes in your hoses causing a leak that’s not really visible until the engine is running.
3.) Since it’s used (New to you) it’s probably a good idea to change your thermostat and the gasket, the gasket is to help prevent any fluid from leaking, the thermostat housing is metal and metal does expand when heated so the gaskets to prevent any leaking. It’s very easy to replace, you can either buy a premade thermostat housing gasket or make your own with the gasket maker that’s specifically for thermostats and water pumps whichever is easiest for you.
4.) if it’s still leaking after replacing any other parts (for example the thermostat) then check to make sure your hoses are on all the way and that the clamps are close enough to the base of the hose. Sometimes the clamp is actually what causes it to leak you can buy cheap ones at most parts stores.
Hope that helps you with your problem.