What is wrong if the starter clicks but won't start if the radio works so it can't be the battery and it has a new starter?

Why CAN'T it be the battery? The radio uses far less electrical energy than does the starter. To KNOW that it's not the battery, use a volt meter and have someone attempt to start the engine while you hold the voltmeter contacts to the battery posts. If the voltage drops below about 10 VDC, it's probably a discharged or faulty battery.

Try jump starting it and see if that solves the problem.

If you can get the engine started, use the voltmeter again and see what the operating voltage is. You should get about 13.5 to 14.5 VDC if your alternator is functioning properly.

If you've isolated all other possibilities, it's probably the starter solenoid. It's integral to the starer so unless you've rebuilt starters in the past, don't bother getting a new one.

But before you just throw replacement parts at the problem, CHECK FIRST! Make sure that's what's really wrong.