The solenoid is an electric fuel enrichment. This explains it better than I can
https://www.buggydepot.com/tech-cent...-how-it-works/
First thing (which is pretty common) on gy6s is check the valve adjustment. Bet 9 out of 10 gy6's I work on that the owner says something is wrong with the carb is valve adjustment issues.
This should be really close to your wiring
http://shahsramblings.com/wp-content...m-download.jpg