A Cold War-style spy saga involving guns, gangsters and the Russian security services is roiling this tiny ex-Soviet state before its election, which has become crucial battleground in the tug of war between Europe and Moscow. Moldova’s election commission on Thursday barred Renato Usatii, a populist pro-Russian candidate, from running in Sunday’s parliamentary elections after a leaked audio recording appeared to show him discussing his close connections to the FSB, the Russian security service and successor to the KGB. Government officials and political leaders here have long alleged that Mr. Usatii is a front for Russian secret services and criminal gangs—part of a multipronged Russian plan to get control over the country, which neighbors Ukraine. The audio recording surfaced as Moldovan police unearthed a cache of weapons and military supplies, including grenade launchers and rifles, in raids on members of a pro-Russian antifascist movement. In Moscow, there was no official comment on the news.