That's the diode rectifier board, it rectifies AC electricity from the alternator, to DC electricity .
The solenoid is on the starter .
You need to remove the top cover cover of the engine to access it .
Before you go in real deep with this problem, the starter relay is under the tank, it has a wiring diagram molded into the top cover, when the starter gives you problems, if you can remove the fuel tank and press down on the top of the cover to manually operate the relay, just to eliminate this part and possibly save you some work .
If this doesn't work, first guess to your starter problem, is the starter is 'dry' on lubricant, it needs a good internal cleaning and re-lubrication and possibly the sintered bushings at each end of the armature shaft .
In the FAQ and Procedures section, there is a pictorial guide on how to disassemble clean and reassemble the starter, including how to clean up the contacts of the solenoid .
Oh yeah, welcome to the forum, post all of the questions you have about your bike, there's a wealth of knowledge here !!!!!
After taking a second look at your photo, I notice a fair amount of corrosion, you may want to check all of the electrical connections for corrosion .
The positive battery cable, goes directly to the starter, we've seen a few examples here, of this cable developing corrosion at the battery end, under the insulation .
