The electrolytic caps in the clock/counter module are failing. I'll guess this is a Taiwan radio. That makes it 33 or more years old. That module is full of digital chips constantly turning each other on and off. They will each draw a major surge of current during those on/off transitions, and the voltage drop from those surges will produce hum/buzz noises in the radio's audio circuits. There are filter caps in the module that provide the current for these surges and smooth them out. Until they go bad.
Electrolytic caps fail from age and use, both. A capacitor like the ones in that radio that was kept on a shelf in the proverbial "cool dry place" will check just fine as a rule. But the heat of the radio's operation will speed up the deterioration. The highest temperatures in that model are found inside the clock/counter module. We see the caps inside that metal shield tend to go bad a year or three before the caps elsewhere in the radio start causing trouble.
Quite frequently the "hum" will exhibit small changes in the sound as the segments in the digits turn on and off when the displayed frequency changes. Switching from clock to counter will generally affect the sound if it's being caused by failed caps on the clock/counter board.
The circuit board inside the clock/counter module is different from the single-sided main pc board for the radio. The counter board has foil traces on both sides, and it has plate-through holes. This means that a tiny copper sleeve in inserted in holes to bridge the top foil to the bottom foil through the hole. If that sleeve is damaged by removing a part, the circuit function it served will quit. Some of the 11 or 12 electrolytic caps you'll need to change have only a "doughnut" foil on the solder side. It will tend to slide off when removing the old capacitor at that spot. Somewhere around here I started to write up the "comb-over" trick we use to get around that. Laying the wire lead down so it reaches to a nearby pad connected by the top-side foil will make sure that cap gets connected securely.
I should find the pictures we shot for that and get it written up.
Makes me wonder if Klondike Mike sells a "counter only" kit of caps for the Cobra 2000. Would be handy.
I could be wrong about the root of your problem, but if unplugging the counter module's power plug shuts up the noise, this should clinch the question of where the noise is coming from.
73