Honda gx390 electric starter

It can be the solenoid but electrical checks are needed to done. Just guessing and replacing suspected parts can get expensive very fast when you may simply have a bad connection. Ground returns can cause the same symptoms as a bad solenoid or starter.
 
This is why shops have battery testers, DVMMs, and should have at least one experienced electrical tech. An inexperienced tech with DVMM is no better off than a tech without one. Sorry to pointed on this but I seen more my fair share of screwed wiring this year from other shops techs just guessing at a problem.
 
A good example of getting expensive fast is the customer that came in last year. He had replace the starter, battery, solenoid, stator, and voltage regulator; all none returnable parts. He finally brought in the mower for me to fix. It turned out to just a bad $0.50 connector at the ignition switch. Yes of course there was shop time billed but he would have not been out all the money for the other parts.