The condition of the battery does not affect how the bike will start on the kick.
Sounds like the voltage is dropping when you hit the start button. If the battery & its connections aren't up to scratch then the great gobs of current the starter draws can drop the voltage when you hit the button to the point where the starter relay won't hold in. It drops out, the starter current stops the voltage rises again to the point where the relay pulls in again. It causes the relay to pull in & drop out so fast it tends to buzz or click as you describe.
Do you have a multimeter? Set it to read volts & connect it across the battery. You should see 8 volts or better while the start button is pressed. If low the battery is either flat or knackered. Do the lights work OK without the engine running, especially while cranking? This may give an idea of battery condition.
If the voltage is OK put your black meter lead on the engine case somewhere & the red lead on the bolt terminal on the starter relay that is connected to the battery. If this you now see less than 7 or 8 volts here while cranking either your battery leads are crook or you have a bad connection on the battery posts or the battery ground lead to the frame. It is remotely possible you have a bad connection between the rear subframe & frame, however the whole back of the bike would be loose!
If this voltage is OK you have a problem somewhere in the start switch circuit, however a fault here is not likely to cause the relay to buzz as you said. It would more likely just not work at all. Put your meter black on the engine case & the red lead to each small wire in turn that go into the start relay, up the butt of the plastic plug. One should never have voltage (ie ground), the other should see better than 7 or 8 volts while cranking. If OK probably the only thing that can be crook is the start relay. If low start checking connections to the keyswitch & start button. Check the start relay wire has not been pinched between the headstock travel stop bolt. This wire may also have fractured inside its insulaton around the headstock where it gets bent when you turn the bars. Check you are getting power to the back of the keyswitch while cranking. It could also be either the keyswitch or start button themselves.
