I think you're in a warm state, at least I got that impression, which makes me think you're not used to this cold snap you're having.
That clicking is the solenoid being activated, after which there's not enough current (oomph from your battery) to run anything else. Your battery has a bench profile of around 50% of normal capacity at 0C than at room temperature. A battery has a steady decline in maximum capacity as it ages. Your battery may only function at 25% of its original power at room temperature. Now that it is cold, its about worthless.
Why can you keep pushing the starter and eventually it can work? As you open (actually close) a big circuit with lots of demand, the battery tries to meet that demand. Electrical flow, if you will, creates heat, not just in wires but in the battery itself, and that heat might be just enough to improve the battery output to the point where it will spin the engine. You said your bike is carbed, if you had the added draw of fuel injection, it would never start with that battery.
Stop trying to start it. Get a new fully charged battery. You can damage the electronics with what you're doing and I don't want to sell you a gauge cluster.
Then, because your bike has problems after a 30 minute ride, check your stator output at the battery by measuring the voltage when the bike is switched off, then again when it is running. You may have a damaged regulator, stator, or connection plug that will lead to over or undercharging the battery, both of which will ruin it.
I'd normally say the first step is to charge your existing battery, but your complaint didn't include the usual "I just got a new battery and now this..." that is the more standard situation.