The starter solenoid is integral to the starter, which is located below the intake manifold and requires a fair amount of disassembly for access.
The start relay (and all relays) consist of a control circuit where a relatively small amount of current (typically less than 200 milli-amps) flows through a fine wire coil around an iron core creating a magnetic field. The magnetic field pulls a thin metal strip towards the magnet, closing circuit contact points through which flows the large amount of output current that the particular relay controls.
When the start relay "buzzes" the thin metal contact strip is not being firmly held in place to the stationary contact by the magnetic field, and the two contact points through which the output current is supposed to flow vibrate against one another - creating noise, an inability for the correct amount of output current to flow (ie: the starter doesn't engage) and electrical arcing (with resultant burning and ultimate damage) between the two contacts.
The real question arises as to why, specifically, the relay contacts are not making fast and secure contact with one another when the relay is energized.
Insufficient current flow through the control coil of the relay will result in a weak magnetic field being genererated. Current flow is directly proportional to available voltage, and inversely proportional to resistance.
The diagnostic of the issue, with resulting "solution" would be to capture electrical data of the event while it is occurring and measuring voltage drops accross various components of the circuit, which would be the ignition switch, related wiring and the relay itself.
Simple concept, and relatively easy diagnosis - provide the diagnostic tests are being performed correctly while the fault is actually present.
Short of proper diagnosis, it's all pure guess-work and speculation.
Disconnect the switched wire to the starter relay and jump the relay with a jumper wire off the battery. I think the switched wire will come from the Aux relay. It should be hot when the starter button is pushed.
The BR color for KTM is usually a ground (-) wire and should show no voltage.
What happens when you jump it, does the starter turn?
Check you connections and plugs especially the neutral switch, clutch switch, key switch. Plug, unplug, clean.
If the nertral switch or clutch switch is bad or the starter button is making poor contact, relays won't help you.