I think the best place to start is to look at your Buffer settings and Plug-In Latency settings. You can find the Buffer settings in Preferences>Devices and the Plug-In Latency settings in Preferences>General.
I'm not sure that either of these will actually resolve your issue but it's a good place to start. What you're describing sounds like the system is starting playback immediately but isn't processing things quickly enough to playback when they're supposed to. This sounds more like a buffering issue to me because the system eventually catches back up to play things at the correct time, implying that the system had the time to buffer those sounds before they came around.
If you're not having success there, here are some additional things that may help people more knowledgable than myself diagnose and resolve your issue. Does this happen with all VSTs (software instruments and plugins) or just this one? Does the track in question have a lot of effects on it? Does it have the same issue when the effects are all turned off? If multiple tracks are playing at once, do they all have this type of latency? Is the latency the exact same amount for every track, or does it vary by track? If it varies, what are the differences between tracks, including which VSTs are being used and how many on each track? Does this just happen with VSTs or does it also happen with audio?
You also need to consider the processing power of your computer. If it's old or on the low end of memory, it may have more to do with your computer than your settings.
If you aren't able to find your answer here, I'd suggest trying the Logic Pro Forum. And for the sake of SE, if you find an answer somewhere else, you should post it as an answer here to allow others in the future to be able to reference it here.